Lecture 1 - Introduction to the problem of regularity Giuseppe Di - - PowerPoint PPT Presentation

REGULARITY FOR ELLIPTIC EQUATIONS UNDER

MINIMAL ASSUMPTIONS

G.Di Fazio

Dipartimento di Matematica e Informatica Universit` a di Catania 13-17th August 2018 – MYSAGA – Bandung – Indonesia

Giuseppe Di Fazio (University of Catania) 1 / 100

Lecture 1

Lecture 1 - Introduction to the problem of regularity

Giuseppe Di Fazio (University of Catania) 2 / 100

Lecture 1

THE EQUATION

We are interested in the generalized solutions of the equation divA(x, u, ∇u) + B(x, u, ∇u) = 0

• n a given bounded domain Ω ⊂ Rn where the measurable functions A

and B satisfy suitable growth assumptions.

Giuseppe Di Fazio (University of Catania) 3 / 100

Lecture 1

THE EQUATION

EXAMPLE

1

A(x, u, ξ) = I · ξ and B = f yield Poisson equation −∆u = f

2

A(x, u, ξ) = I · |ξ|p−2ξ and B = f yield p-Laplace equation −∆pu = f, 1 < p < ∞

Giuseppe Di Fazio (University of Catania) 4 / 100

Lecture 1

SOLUTIONS

What do we mean by solution of a PDE ?

1

Classical (smooth data)

2

Weak (discontinuous data)

3

Very weak (measure data )

Giuseppe Di Fazio (University of Catania) 5 / 100

Lecture 1

SOLUTIONS

divA(x, u, ∇u) + B(x, u, ∇u) = 0 where the data may be smooth functions, discontinuous functions or even measures or distributions.

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Lecture 1

LINEAR EQUATIONS

In the linear case A and B are linear with respect to the u and ∇u arguments. Lu ≡ −div(A(x)∇u) + V(x)u + b(x) · ∇u = f(x) + divg(x) We will assume that A is a symmetric positive definite matrix of bounded measurable functions in Ω. Moreover, we assume that there exists ν > 0 such that A(x)ξ, ξ ≥ ν|ξ|2 for a.e. x ∈ Ω and any ξ ∈ Rn.

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Lecture 1

LINEAR EQUATIONS

Moreover - to keep the things as simple as possible - we set V = 0 b = g = 0 and give the definition of weak solution.

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Lecture 1

WEAK SOLUTIONS

DEFINITION Given a distribution f in W −1,2(Ω), a function u in W 1,2(Ω) is a solution

• f the equation

−div(A(x)∇u) = f in Ω if ˆ

Ω

A(x)∇u∇ϕ dx + ˆ

Ω

f(x)ϕ dx = 0 ∀ϕ ∈ W 1,2 (Ω)

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Lecture 1

WEAK SOLUTIONS - CLASSICAL RESULTS

We recall the outstanding regularity result by De Giorgi - Nash - Moser Theorem for linear homogeneous elliptic equations, i.e. V = b = f = g = 0. THEOREM Any weak solution of the equation Lu = 0 is α h¨

• lder continuous in Ω.

The exponent α is a function of n and ν only. Moreover, there exists a constant c = c(n, ν) such that the following estimate hold true

• sc

BR(y) u ≤ c

R R0 α

BR0(y)

|u|2 dx 1/2 ∀R < R0 .

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Lecture 1

WEAK SOLUTIONS - CLASSICAL RESULTS

De Giorgi’s result was extended by several Authors to the linear uniformly elliptic equation Lu ≡ −div(A(x)∇u) + V(x)u + b · ∇u = f(x) + divg(x) under suitable integrability of the functions V, b, f and g. In general, the assumptions are on how large the integrability exponent is in such a way to ensure smoothness of the weak solutions.

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Lecture 1

WEAK SOLUTIONS - CLASSICAL RESULTS

No smoothness is required to the coefficients A(x)

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Lecture 1

OPTIMALITY OF THE EXPONENTS

The lower the integrability exponent is, the better and general the result. EXAMPLE The function u(x) = log | log |x|| ∀0 < |x| < R for suitably small R is a W 1,2 function that is a solution of ∆u = n − 2 |x|2 log |x| − 1 |x|2 log2 |x| RHS belongs to Lp for p < n/2 The solution is not locally bounded and then NO REGULARITY.

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Lecture 1

OPTIMALITY OF THE EXPONENTS

EXAMPLE The functions u(x) and v(x) = log | log R| for suitable R < 1 are both W 1,2 functions solutions of the Dirichlet problem    ∆u = n − 2 |x|2 log |x| − 1 |x|2 log2 |x| 0 < |x| < R u = log | log R| |x| = R RHS belongs to Lp for p < n/2 The Dirichlet problem is not well posed.

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Lecture 1

OPTIMALITY OF THE EXPONENTS

Previous examples show that there exists an integrability threshold that is a function of dimension and this phenomena does not depend on how bad are the coefficients A(x).

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Lecture 1

OPTIMALITY OF THE EXPONENTS

Classical regularity results (Stampacchia - Ladyzhenskaya - Uraltzeva .... ) show that the solution is smooth if (and only if) the coefficients have integrability more than the threshold.

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Lecture 1

OPTIMALITY OF THE EXPONENTS

Large exponent of integrability implies regularity of the solutions. But this is not necessary!

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Lecture 1

NEW POINT OF VIEW IN THE 80’S

Apparently, the examples above give no hope for regularity below the threshold. In 1982 Aizenman & Simon showed - using probabilistic arguments - that the weak solutions of the equation ∆u + Vu = 0 are continuous. The assumption regarding V is the following

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Lecture 1

NEW POINT OF VIEW IN THE 80’S

DEFINITION (STUMMEL-KATO) Let V be a locally integrable function in Rn. Consider φ(r) = sup

x∈Rn

ˆ

Ω∩B(x,r)

|V(y)| |x − y|n−2 dy . We say that V belongs to the Stummel - Kato classes ˜ S or S if the function φ is bounded or infinitesimal in a neighborhood of zero

• respectively. The function φ is called the Stummel - Kato modulus of
• V. If Ω is a domain we say V ∈ S(Ω) or V ∈ ˜

S(Ω) means V(x)χΩ(x) belongs to S or ˜ S respectively.

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Lecture 1

PROGRESS FROM 80’S TO 90’S

In 1986 Chiarenza, Fabes & Garofalo gave a different proof of Aizenman & Simon result by analytical proof true for the uniformly elliptic equation div(A(x)∇u) + Vu = 0 assuming V in the Stummel - Kato class S(Ω).

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Lecture 1

STUMMEL - KATO IMPLIES REGULARITY

THEOREM (CHIARENZA FABES GAROFALO) Let u be a weak solution of the equation div(A(x)∇u) + Vu = 0 in Ω ⊂ Rn, n ≥ 3. Let us assume V in the Stummel - Kato class S(Ω). Then, any weak solution u is continuous in Ω.

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Lecture 2

Lecture 2 - Very weak solution and Green function

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Lecture 2

STUMMEL - KATO AND LEBESGUE

1

The assumption V ∈ S(Ω) is much more general than the previous V ∈ Lp(Ω) where p > n/2.

2

It does not contradict the examples.

3

It says that Lebesgue spaces are not the right assumptions for regularity.

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Lecture 2

MORREY SPACES

To better understand Stummel - Kato class we introduce another family

• f function spaces.

DEFINITION (MORREY SPACES) Let f be a locally integrable function in Ω ⊂ Rn, 1 ≤ p < ∞, 0 < λ < n. We say that f belongs to the Morrey space Lp,λ(Ω) if fp

p,λ ≡

sup

x∈Ω,0<r

1 r λ ˆ

Ω∩B(x,r)

|f(y)|p dy < ∞

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Lecture 2

COMPARISON

THEOREM The following inclusion hold true Lp ⊂ S(Ω) p > n/2 and L1,λ ⊂ S n − 2 < λ < n with proper inclusions.

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Lecture 2

COMPARISON

PROOF. By H¨

• lder inequality

ˆ

B(x,r)

|f(y)| |x − y|n−2 dy ≤ fp ˆ

B(x,r)

|x − y|(n−2)p′ dy 1/p′ and - if p > n/2 - the RHS is finite. Namely we get φf(r) ≤ fpr 2−n/p

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Lecture 2

COMPARISON

PROOF. By H¨

• lder inequality

ˆ

B(x,r)

|f(y)| |x − y|n−2 dy =

+∞

• k=0

ˆ

B(x,r/2k)\B(x,r/2k+1)

|f(y)||x − y|2−n dy ≤ f1,λ

+∞

• k=0

r 2k λ−n+2 and - if λ > n − 2 - RHS is finite. Namely we get φf(r) ≤ cf1,λr λ−n+2

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Lecture 2

A NEW SCALE THAT GIVES REGULARITY

A natural question then is Can we obtain better regularity results assuming the coefficients in Morrey spaces?

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Lecture 2

BAD NEWS

Morrey spaces are not so easy to handle as Lebesgue spaces are.

1

Morrey spaces are not the closure of smooth functions

2

Mollifiers do not converge

3

The Morrey norm has not the Absolute continuity property i.e. uLp,λ(BR) is not infinitesimal with R.

4

For any 1 ≤ p < ∞ there exists a function u in Lp,λ such that u / ∈ Lq for any q > p.

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Lecture 2

GOOD NEWS

1

Morrey spaces imply regularity

2

If the RHS has a sign they are necessary for regularity. Morrey spaces are necessary and sufficient for regularity

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Lecture 2

NECESSITY OF VERY WEAK SOLUTION

1

In general Lp,λ is not contained in W −1,2

2

This means that - in general - weak solutions do not exist

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Lecture 2

VERY WEAK SOLUTION

This force us to introduce the concept of very weak solution

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Lecture 2

VERY WEAK SOLUTION

DEFINITION (VERY WEAK SOLUTION) Let Ω be a bounded domain in Rn and let µ be a bounded variation measure in Ω. A function u ∈ L1(Ω) is a very weak solution of the Dirichlet problem

• Lu = µ

in Ω u = 0

• n ∂Ω

if, for any ϕ ∈ W 1,2 (Ω) ∩ C0(Ω) such that Lϕ ∈ C0(Ω), we have ˆ

Ω

uLϕ dx = ˆ

Ω

ϕdµ.

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Lecture 2

VERY WEAK SOLUTION

REMARK The class of test functions is non empty by De Giorgi regularity Theorem. Indeed, let ψ be a C0(Ω) given function. Then, there exists a h¨

• lder

continuous weak solution ϕ of the equation Lϕ = ψ so ϕ ∈ W 1,2 (Ω) and then ϕ ∈ C0(Ω) ∩ W 1,2 (Ω).

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Lecture 2

VERY WEAK SOLUTION

REMARK Very weak solution is unique. Indeed, the homogeneous problem

• Lu = 0

in Ω u = 0

• n ∂Ω

and show that u = 0. Let ψ ∈ C0(Ω) and ϕ in W 1,2 (Ω) ∩ C0(Ω) such that Lϕ = ψ. Since ˆ

Ω

u Lϕ dx = 0 i.e. ˆ

Ω

u ψ dx = 0 for any continuous ψ, we have u = 0 in Ω.

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Lecture 2

GREEN FUNCTION

Very important for linear differential operators is the Green function.

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Lecture 2

GREEN FUNCTION

Let us consider the problem

• Lu = T

in Ω u = 0

• n ∂Ω

By the definition of very weak solution, there exists a linear application G : W −1,2(Ω) → W 1,2 (Ω) defined by G(T) = u This is what we call the Green operator.

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Lecture 2

GREEN FUNCTION

Now, by the local boundedness and the local h¨

• lder continuity

Theorems we have G : W −1,2(Ω) → W 1,2 (Ω) such that for any T ∈ W −1,2(Ω) the function u = G(T) is the unique weak solution in W 1,2 (Ω) of the Dirichlet problem.

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Lecture 2

GREEN FUNCTION

THEOREM For any bounded variation measure µ on Ω there exists a unique solution of the equation Lu = µ that is zero on the boundary ∂Ω. Moreover it belongs to W 1,p′ (Ω) for any p > n.

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Lecture 2

GREEN FUNCTION

The operator G maps continuously W −1,p(Ω) in C0(Ω) Indeed, if p > n, by De Giorgi - Nash - Moser Theorem we have G : W −1,p(Ω) → C0(Ω) and there exists c such that max

¯ Ω |G(ψ)| ≤ c ψ−1,p

∀ψ ∈ W −1,p(Ω)

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Lecture 2

GREEN FUNCTION

Then, u is a very weak solution of Lu = µ vanishing on ∂Ω if and only if ˆ

Ω

uψ dx = ˆ

Ω

G(ψ) dµ for all ϕ ∈ C0(Ω) and

• ˆ

Ω

uψ dx

• =
• ˆ

Ω

G(ψ) dµ

• ≤ c

ˆ

Ω

|dµ|ψ−1,p for all ϕ ∈ C0(Ω).

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Lecture 2

GREEN FUNCTION

By density we have uW 1,p′ ≤ c ˆ

Ω

|dµ| for p > n. The application µ → u is the adjoint of G, i.e. u = G∗(µ).

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Lecture 2

GREEN FUNCTION

Since G : W −1,p → C0(Ω) is continuous by duality we have that G∗ is also continuous from the space M of the measures with bounded variation in Ω to W 1,p′ (Ω). For any µ ∈ M we have G∗(µ) ∈ W 1,p′ (Ω).

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Lecture 2

WEAK AND VERY WEAK

Now we are ready to compare the notions of weak and very weak solutions. THEOREM (LITTMAN - STAMPACCHIA - WEINBERGER) Let Ω ⊂ Rn be a bounded domain and µ be a bounded variation

• measure. Let u ∈ L1(Ω) be the very weak solution of the Dirichlet

problem

• Lu = µ

in Ω u = 0

• n ∂Ω.

Then u is a weak solution if and only if µ ∈ W −1,2(Ω).

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Lecture 2

WEAK AND VERY WEAK

PROOF. Let u be the unique weak solution in W 1,2 (Ω) of equation Lu = µ. We have ˆ

Ω

aijuxiφxj dx = ˆ

Ω

φ dµ for any φ ∈ W 1,2 (Ω) and then,

• ˆ

Ω

φ dµ

• ≤ ν∇u2∇φ2

for all φ ∈ W 1,2 (Ω), which means that µ ∈ W −1,2(Ω).

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Lecture 2

WEAK AND VERY WEAK

PROOF. Now, if µ ∈ W −1,2(Ω) there exists f such that µ = divf and then the equation Lu = divf has a weak solution by classical Hilbert space approach.

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Lecture 2

GREEN FUNCTION

DEFINITION (GREEN FUNCTION) If y ∈ Ω the Dirac mass at y, δy is a bounded variation measure. Then we may consider the very weak solution g(·, y) of the Dirichlet problem in the case µ ≡ δy. Such a function will be called the Green function for the operator L with respect to the domain Ω with pole at y.

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Lecture 2

GREEN FUNCTION

REMARK The Green function satisfies ˆ

Ω

g(x, y)Lϕ(x)dx = ˆ

Ω

ϕ(x)dδy(x) for all ϕ ∈ C0(Ω) ∩ W 1,2 (Ω) such that Lϕ ∈ C0(Ω). Then, by definition

• f δy, we have

ϕ(y) = ˆ

Ω

g(x, y)Lϕ(x)dx .

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Lecture 2

GREEN FUNCTION

Using the Green function we may represent the weak solution of the Dirichlet problem. Indeed, let ψ ∈ C0(Ω) and ϕ be the weak solution of

• Lϕ = ψ

in Ω ϕ = 0

• n ∂Ω.

Since C0(Ω) ⊂ W −1,2(Ω) we have ϕ ∈ W 1,2 (Ω) and, by De Giorgi Theorem, ϕ ∈ C0(Ω). This implies ˆ

Ω

g(x, y)ψ(x)dx = ˆ

Ω

ϕ(x)dδy(x) and then ϕ(y) = ˆ

Ω

g(x, y)ψ(x)dx .

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Lecture 2

GREEN FUNCTION PROPERTIES

THEOREM (GR ¨

UTER & WIDMAN)

There exists a unique function G : Ω × Ω → R ∪ {∞} such that

• 1. G(x, y) ≥ 0 where it is defined.
• 2. For any y ∈ Ω and any r > 0 such that Br(y) ⊂ Ω the function

G(·, y) belongs to W 1,2(Ω \ Br(y)) ∩ W 1,1 (Ω).

• 3. The following relation holds true

ˆ

Ω

aij(x)Gxi(x, y)ϕxj(x) dx = ϕ(y) for all ϕ ∈ C∞

0 (Ω).

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Lecture 2

PROPERTIES OF GREEN FUNCTION

THEOREM Moreover, if we set G(x) ≡ G(x, y), the function G satisfies the following properties

• 4. G belongs to the space Ln/(n−2)

w

(Ω) with bounds depending on the ellipticity and dimension only.

• 5. ∇G belongs to the space Ln/(n−1)

w

(Ω) with bounds depending on the ellipticity and dimension only.

• 6. G belongs to the space W 1,s

0 (Ω) for any 1 ≤ s <

n n − 1 with bounds depending on the ellipticity, dimension and the exponent s

• nly.

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Lecture 2

PROPERTIES OF GREEN FUNCTION

THEOREM

• 7. There exists a positive constant c depending on the ellipticity and

dimension only such that G(x, y) ≤ c|x − y|2−n for all x, y ∈ Ω , x = y.

• 8. There exists a positive constant c depending on the ellipticity and

dimension only such that G(x, y) ≥ c|x − y|2−n for all x, y ∈ Ω such that 0 < |x − y| ≤ 1 2d(y, ∂Ω) , x = y.

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Lecture 2

ONLY ONE GREEN FUNCTION

THEOREM The Green function g defined by Stampacchia and the other one G defined by Gr¨ uter & Widman are the same function.

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Lecture 2

REPRESENTATION FORMULA

We can represent the very weak solution of the Dirichlet problem

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Lecture 2

REPRESENTATION FORMULA

THEOREM (REPRESENTATION FORMULA) Let µ be a bounded variation measure in a bounded domain Ω ⊂ Rn(n ≥ 3) and let u ∈ L1(Ω) be the very weak solution of the Dirichlet problem

• Lu = µ

in Ω u = 0

• n ∂Ω.

Then, the following representation formula holds true u(x) = ˆ

Ω

g(x, y)dµ(y) where g(x, y) is the Green’s function for the operator L with respect to Ω with pole at y ∈ Ω.

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Lecture 2

REPRESENTATION FORMULA

PROOF. We know existence and uniqueness. The proof by direct substitution. Let ϕ ∈ W 1,2 (Ω) ∩ C0(Ω) be such that Lϕ ∈ C0(Ω). Then ˆ

Ω

ϕ(y)dµ(y) = ˆ

Ω

ˆ

Ω

g(x, y)Lϕ(x)dx

• dµ(y)

= ˆ

Ω

ˆ

Ω

g(x, y)dµ(y)

• Lϕ(x)dx

= ˆ

Ω

u(x)Lϕ(x)dx and then ˆ

Ω

u(x)Lϕ(x)dx = ˆ

Ω

ϕ(x)dµ(x) for any ϕ ∈ W 1,2 (Ω) ∩ C0(Ω) such that Lϕ ∈ C0(Ω) that is the result.

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Lecture 2

REGULARITY

Representation formula will give us important information about the REGULARITY

• f the very weak solution.

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Lecture 3

Lecture 3 - Sufficient conditions for regularity

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Lecture 3

BUN BUN I will not go to your boring class! I’m on holiday!

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Lecture 3

REGULARITY OF VERY WEAK SOLUTION

THEOREM Let 0 < λ < n − 2, f ∈ L1,λ(Ω) and let u be the very weak solution of the Dirichlet problem

• Lu = f

in Ω u = 0

• n ∂Ω

Then, u ∈ Lpλ,λ

w

(Ω) where 1 pλ = 1 − 2 n − λ . In particular, u ∈ Lp(Ω) for any 1 ≤ p < pλ. Moreover, there exists c ≥ 0 such that uLp ≤ c fL1,λ where c does not depend on u and f.

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Lecture 3

REGULARITY OF VERY WEAK SOLUTION

The proof easily follows from Chiarenza – Frasca Theorem.

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Lecture 3

CHIARENZA – FRASCA THEOREM

THEOREM (CHIARENZA – FRASCA) Let 1 < p < +∞ and 0 < λ < n. Then, there exists a constant c which depend on n, p and λ such that MfLp,λ ≤ cfLp,λ . If p = 1 we have the following weak type estimate t| {y ∈ Br(x) : Mf(y) > t} | ≤ c r λfL1,λ . For 1 ≤ p ≤ +∞, 0 < λ < n the function Mf is finite for almost all x ∈ Rn.

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. By Representation Formula we have |u(x)| ≤ ˆ

Ω

g(x, y) |f(y)|dy ≤ c ˆ

Ω

|x − y|2−n|f(y)|dy a.e. in Ω where g(x, y) is the Green’s function of L with respect to Ω with pole at y ∈ Ω and c is a constant which depends on n and ν.

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. Since Br(x) ⊂ B2r(x) ⊂ Ω we have ˆ

Ω

|f(y)| |x − y|n−2 dy = ˆ

B2r(x)

|f(y)| |x − y|n−2 dy+ + ˆ

Ω\B2r(x)

|f(y)| |x − y|n−2 dy ≡ I + II.

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. We separately estimate the two integrals by using

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. We have I =

∞

• k=0

ˆ

B(x;r/2k−1)\B(x;r/2k)

|f(y)| |x − y|n−2 dy ≤ ≤ c

∞

• k=0

r 2k 2 Mf(x) = c r 2Mf(x) where Mf(x) is the Hardy-Littlewood maximal function of f at x ∈ Ω.

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. Now estimate II. We have II =

∞

• k=1

ˆ

r2k+1≤|x−y|<r2k

|f(y)| |x − y|n−2 dy ≤ ≤ c

∞

• k=0
• 2kr

λ−n+2 f1,λ = c r λ−n+2f1,λ .

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. Then, for any r > 0, ˆ

Ω

|f(y)| |x − y|n−2 dy ≤ Cn,λψ(r) where ψ(r) ≡ r 2Mf(x) + r λ−n+2f1,λ−n+2 for r > 0.

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. By taking the minimum of the right hand side we get ˆ

Ω

|f(y)| |x − y|n−2 dy ≤ c (Mf(x))1/pλ f

2 n−λ

1,λ

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Lecture 3

PROOF OF REGULARITY RESULT

PROOF. so that |u(x)| ≤ c ˆ

Ω

|f(y)| |x − y|n−2 dy ≤ cf

2 n−λ

1,λ (Mf(x))1/pλ

a.e. in Ω and the result follows by Chiarenza - Frasca Theorem.

Giuseppe Di Fazio (University of Catania) 63 / 100

Lecture 3

A COUNTEREXAMPLE

If λ → n − 2 then pλ → ∞. Unfortunately the implication f ∈ L1,n−2(Ω) ⇒ u ∈ L∞(Ω) is not true! Let Ω = {x ∈ Rn : 0 < |x| < 1}, n ≥ 3. Let us check that the very weak solution of the Dirichlet problem    ∆u = n − 2 |x|2 in Ω u = 0

• n ∂Ω

is the function u(x) = log |x|.

Giuseppe Di Fazio (University of Catania) 64 / 100

Lecture 3

VERY WEAK SOLUTION IN BMO

THEOREM If f ∈ L1,n−2(Ω) the very weak solution u belongs to BMO locally in the following sense. Let Ω′ ⋐ Ω and d = dist(Ω′, ∂Ω). Then, there exists a constant C ≡ C(n, ν, d) > 0 such that

Br(x)

• u(y) − uBr(x)
• dy ≤ Cf1,n−2 .

for all 0 < r < d 2 and x ∈ Ω′.

Giuseppe Di Fazio (University of Catania) 65 / 100

Lecture 3

VERY WEAK SOLUTION = WEAK SOLUTION

THEOREM ˜ S(Ω) ⊂ W −1,2(Ω) PROOF. Let f ∈ ˜ S(Ω) and ϕ ∈ C∞

0 (Ω).

We have to show that there exists a positive constant C such that |f, ϕ| ≤ Cφ(f)∇ϕ2 Set I1(f)(y) = ´

Ω |f(x)| |x−y|n−1 dx. Then

|f, ϕ| ≤ C ˆ

Ω

|f(x)| ˆ

Ω

|∇ϕ(y)| |x − y|n−1 dy

• dx

= C ˆ

Ω

|∇ϕ(y)|I1(f)(y) dy ≤ C∇ϕL2 I1(f)L2 .

Giuseppe Di Fazio (University of Catania) 66 / 100

Lecture 3

VERY WEAK SOLUTION = WEAK SOLUTION

THEOREM ˜ S(Ω) ⊂ W −1,2(Ω) PROOF. But I1(f)2 = ˆ

Ω

ˆ

Ω

|f(x)| |x − y|n−1 dx

• ˆ

Ω

|f(z)| |z − y|n−1 dz

• dy

= ˆ

Ω

|f(z)| ˆ

Ω

|f(x)| ˆ

Ω

dy |x − y|n−1|z − y|n−1

• dx
• dz.

Giuseppe Di Fazio (University of Catania) 66 / 100

Lecture 3

VERY WEAK SOLUTION = WEAK SOLUTION

THEOREM ˜ S(Ω) ⊂ W −1,2(Ω) PROOF. By well known properties of Riesz potentials we get I1(f)2

L2(Ω) ≤ C

ˆ

Ω

|f(z)| ˆ

Ω

|f(x)| |x − z|n−2 dx dz. Since f ∈ ˜ S(Ω), we have I1(f)2

L2(Ω) ≤ C

ˆ

Ω

|f(x)| dx < ∞ Thus, the conclusion follows putting together the previous inequalities.

Giuseppe Di Fazio (University of Catania) 66 / 100

Lecture 3

VERY WEAK SOLUTION = BOUNDED WEAK SOLUTION

THEOREM If f ∈ ˜ S(Ω) then the (weak) solution u is bounded in Ω. PROOF. For any x ∈ Ω we have |u(x)| ≤ ˆ

Ω

g(x, y)|f(y)|dy ≤ c ˆ

Ω

|f(y)||x − y|2−ndy ≤ c sup

r>0

ˆ

Ω∩Br(x)

|f(y)||x − y|2−ndy ≤ c sup

r>0 x∈Ω

ˆ

Ω∩Br(x)

|f(y)||x − y|2−ndy .

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Lecture 3

WEAK SOLUTION IS CONTINUOUS

THEOREM (CHIARENZA – FABES – GAROFALO) If f ∈ S(Ω), then any weak solution u of equation Lu = f is continuous in Ω.

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Lecture 3

USEFUL INEQUALIES

THEOREM (CACCIOPPOLI) Let u be a weak solution of equation Lu = 0. Then there exists a constant c = c(n, ν) such that ˆ

Ω

|∇u|2ϕ2 dx ≤ ˆ

Ω

u2|∇ϕ|2 dx ∀ϕ ∈ D(Ω) . THEOREM (HARNACK) Let u be a non negative weak solution of equation Lu = 0. Then there exists a constant c = c(n, ν) such that, for any ball B such that 2B ⋐ Ω we have sup

B

u ≤ c inf

B u

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Lecture 3

PROOF. Let η be the Stummel modulus of f. By the embedding the solution is weak and bounded. We have ˆ

Ω

A(x)∇u∇ψdx = ˆ

Ω

f(x)ψ(x)dx for all ψ ∈ C∞

0 (Ω). If Br is a ball such that B4r ⋐ Ω let φ be a cut-off

function C∞

0 (Ω) such that 0 ≤ φ ≤ 1 in Ω, φ ≡ 1 in B3r/2, φ ≡ 0 out of

B2r.

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Lecture 3

PROOF. Then uφ is a weak solutions of L(uφ) = fφ − div (A(x)u∇φ) − A(x)∇u∇φ .

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. By representation formula we get u(x)φ(x) = ˆ

Ω

f(y)φ(y)g(x, y)dy+ + ˆ

Ω

∇yg(x, y)A(y)u(y)∇φ(y)dy − ˆ

Ω

∇u(y)A(y)∇φ(y)g(x, y)dy .

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. For any x ∈ Br/2(x0) u(x) − u(x0) = ˆ

Ω

f(y)φ(y) (g(x, y) − g(x0, y)) dy − ˆ

Ω

A(y)∇u∇φ (g(x, y) − g(x0, y)) dy + ˆ

Ω

• ∇gy(x, y) − ∇gy(x0, y)
• A(y)u(y)∇φdy

≡ I + II + III

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. First estimate I. Let N > 1 to be chosen later. |I| ≤ ˆ

{y∈Ω:|x0−y|>N|x−x0|}

|f(y)φ(y) (g(x, y) − g(x0, y))| dy + ˆ

{y∈Ω:|x0−y|≤N|x−x0|}

|f(y)φ(y) (g(x, y) − g(x0, y))| dy ≡ A + B. To estimate A we use the fact that the Green’s function g(·, y) is α-H¨

• lder continuous out of the pole because of the De Giorgi Theorem

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. Namely, the following inequality holds true |g(x, y) − g(x0, y)| ≤ C |x0 − x| r α

Br

|g(x, y)|2dx 1

2

≤ CN−α max

x∈Br g(x, y)

≤ CN−α min

x∈Br g(x, y)

≤ CN−αg(x0, y) ≤ CN−α|x0 − y|2−n

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. and then A ≤ CN−α ˆ

B2r(x0)

|f(y)|φ(y)|x0 − y|2−ndy ≤ CN−αη(2r).

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. Now estimate B by using Gr¨ uter & Widman Theorem. |g(x, y) − g(x0, y)| ≤ C

• 1

|x − y|n−2 + 1 |x0 − y|n−2

• Giuseppe Di Fazio (University of Catania)

70 / 100

Lecture 3

PROOF. Then, due to the domain of integration, B ≤ C ˆ

|x0−y|≤N|x−x0|

|f(y)| |x − y|n−2 dy + ˆ

|x0−y|≤N|x−x0|

|f(y)| |x0 − y|n−2 dy ≤ C ˆ

|x−y|≤(N+1)|x−x0|

|f(y)| |x − y|n−2 dy + η(N|x − x0|) ≤ ≤ Cη((N + 1)|x0 − x|) + η(N|x − x0|)

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. By choosing now N =

• r

|x − x0| 1

2

we get |I| ≤ |x − x0| r α/2 η(2r)+ + η(

• r|x − x0|) + η(
• r|x − x0| + |x − x0|) .

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. Now we estimate II and III. II = ˆ

B2r\B3r/2

(g(x, y) − g(x0, y))A(y)∇u∇ϕdy By De Giorgi Theorem there exists α ≡ α(n, ν) > 0 such that |g(x, y) − g(x0, y)| ≤ c |x − x0| r α 1 |x0 − y|n−2 if y ∈ B2r \ B3r/2, so that

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. |II| ≤ c r |x − x0| r α ˆ

B2r\B3r/2

|∇u| |x0 − y|n−2 dy ≤ cr 1−n |x − x0| r α ˆ

B2r

|∇u|dy and then |II| ≤ c(n, ν) |x − x0| r α r

• B2r

|∇u|2dy 1

2

.

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. Then, by Caccioppoli inequality |II| ≤ c(n, ν) |x − x0| r α

B4r

|u|2dy 1

2

.

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. Finally we estimate III. III = ˆ

B2r\B3r/2

• ∇gy(x, y) − ∇gy(x0, y)
• A(y)∇ϕu(y)dy

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. By Cauchy Schwarz inequality and Caccioppoli inequality we have |III| ≤ c r ˆ

B2r\B3r/2

|∇gy(x, y) − ∇gy(x0, y)||u|dy ≤ c r ˆ

B2r

|u|2dy 1

2

ˆ

B2r\B3r/2

|∇gy(x, y) − ∇gy(x0, y)|2dy 1

2

= c r 2 ˆ

B2r

|u|2dy 1

2

• ˆ

3 4 r<|x0−y|< 9 4r

|g(x, y) − g(x0, y)|2dy 1

2 Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. De Giorgi Theorem and pointwise estimates of Green function yield |III| ≤ c |x − x0| r α

B2r

|u|2dy 1

2 Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

PROOF. Merging previous estimates we get |u(x) − u(x0)| ≤ c

• η(2r)

|x − x0| r α/2 + η(

• r|x − x0|)

+ η(

• r|x − x0| + |x − x0|)

+ |x − x0| r α

B2r

|u|2dy 1

2

• → 0

as x → x0 because f ∈ S(Ω).

Giuseppe Di Fazio (University of Catania) 70 / 100

Lecture 3

WEAK SOLUTION IS H ¨

OLDER CONTINUOUS

THEOREM If f ∈ L1,λ(Ω), n − 2 < λ < n, then any weak solution u of Lu = f belong to C0,α(Ω) where α ≡ α(n, λ, ν, f1,λ).

Giuseppe Di Fazio (University of Catania) 71 / 100

Lecture 3

PROOF. It is a refinement of the previous result because L1,λ is contained in S. In order to show the result we use the fact that the function f belongs to the Morrey space L1,λ with n − 2 < λ < n that implies the following estimate η(r) ≤ c f1,λr λ−n+2 .

Giuseppe Di Fazio (University of Catania) 72 / 100

Lecture 3

PROOF. By using the estimate we finally get |u(x) − u(x0)| ≤ c f1,λ |x − x0| r α/2 r λ−n+2+ +

• r|x − x0| + |x − x0|

λ−n+2 + c |x − x0| r α

B2r

|u|2dy 1

2 Giuseppe Di Fazio (University of Catania) 72 / 100

Lecture 3

PROOF. and then |u(x) − u(x0)| ≤ c|x − x0|β where β = 1 2 min (λ − n + 2, α) where α is the H¨

• lder exponent of the elliptic operator L arising from

De Giorgi Theorem.

Giuseppe Di Fazio (University of Catania) 72 / 100

Lecture 4

NECESSARY CONDITIONS FOR REGULARITY

Lecture 4 - Necessary conditions for regularity

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Lecture 4

GOAL

Reverse the implication proven before.

Giuseppe Di Fazio (University of Catania) 74 / 100

Lecture 4

NECESSARY CONDITIONS

If u is a very weak solution of the problem

• Lu = f

in Ω u = 0

• n ∂Ω

that enjoys some given regularity properties, which conditions the function f must satisfy?

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Lecture 4

OPEN PROBLEM

In general this problem is still open. The problem has been solved only in a special case.

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Lecture 4

NECESSARY CONDITIONS

We prove some results in this direction under the extra assumption f ≥ 0. We start with the result concerning Lp regularity.

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Lecture 4

SCHECHTER SPACES

DEFINITION Let 1 ≤ p ≤ ∞. The Schechter spaces Mp is the set of all functions f ∈ L1(Ω) for which there exists δ > 0 such that Mp,δ(f) ≡        ´

Ω

´

Ω∩Bδ(x) |f(y)| |x−y|n−2 dy

p dx 1

p

if p < ∞ sup

x∈Ω

´

Ω∩Bδ(x) |f(y)| |x−y|n−2 dy,

if p = ∞ is finite. Local versions are defined as usual.

Giuseppe Di Fazio (University of Catania) 78 / 100

Lecture 4

INCLUSION RELATION AMONG SCHECHTER SPACES

THEOREM Let Ω be a bounded domain in Rn, n ≥ 3 and 1 ≤ p < q ≤ ∞ Then, M∞ ⊆ Mq ⊆ Mp

Giuseppe Di Fazio (University of Catania) 79 / 100

Lecture 4

INTEGRABILITY OF VERY WEAK SOLUTION

THEOREM Let f ∈ L1(Ω), f ≥ 0, 1 < p < ∞ and u ∈ L1(Ω) be the very weak solution of the problem

• Lu = f

in Ω u = 0

• n ∂Ω.

Then, u ∈ Lp

loc(Ω) if and only if f ∈ Mp loc(Ω).

Giuseppe Di Fazio (University of Catania) 80 / 100

Lecture 4

PROOF

PROOF. Let K be a compact subset of Ω. Then, if u ∈ Lp(K) by representation formula and the positivity of f we have ˆ

K

|u(x)|pdx = ˆ

K

ˆ

Ω

g(x, y)f(y)dy p dx ≥ ˆ

K

ˆ

{y∈Ω:|x−y|≥δ}

g(x, y)f(y)dy p dx ≡ ˆ

K

|uδ(x)|pdx

Giuseppe Di Fazio (University of Catania) 81 / 100

Lecture 4

PROOF

PROOF. where uδ(x) ≡ ˆ

{y∈Ω:|x−y|≥δ}

g(x, y)f(y)dy , δ > 0 . We have 0 ≤ uδ(x) ≤ u(x) for any δ > 0 and almost all x ∈ K. Thus, 0 ≤ u(x) − uδ(x) ≤ u(x) ∈ Lp(K) and then lim

δ→0+

ˆ

K

|uδ(x) − u(x)|pdx = 0

Giuseppe Di Fazio (University of Catania) 81 / 100

Lecture 4

PROOF

PROOF. that is lim

δ→0+

ˆ

K

ˆ

{y∈Ω:|x−y|<δ}

g(x, y)f(y)dy p dx = 0 . Now choose δ < 1 2dist(K, ∂Ω) and apply Green’s function estimates to

• btain

lim

δ→0+

ˆ

K

ˆ

{y∈Ω:|x−y|<δ}

|x − y|2−nf(y)dy p dx = 0 . Thus, f ∈ Mp

loc(Ω).

Giuseppe Di Fazio (University of Catania) 81 / 100

Lecture 4

PROOF

PROOF. Let us assume now that f ∈ Mp

loc(Ω) and show that u ∈ Lp loc(Ω). By

representation formula we have u(x) = ˆ

{y∈Ω:|x−y|<δ}

f(y) |x − y|n−2 dy if we choose δ ≥ diamΩ. Meanwhile the integral belongs to Lp(K) for any K compact subset in Ω by the definition of Scheter class Mp

loc(Ω).

Thus u ∈ Lp(K) for all compact sets K.

Giuseppe Di Fazio (University of Catania) 81 / 100

Lecture 4

CONDITION FOR BOUNDEDNESS

THEOREM Let f ∈ L1(Ω), f ≥ 0 and u ∈ L1(Ω) be the very weak solution of the problem

• Lu = f

in Ω u = 0

• n ∂Ω.

Then, u ∈ L∞

loc(Ω) if and only if f ∈ ˜

Sloc(Ω).

Giuseppe Di Fazio (University of Catania) 82 / 100

Lecture 4

PROOF. We already know the condition is sufficient. We assume u to be locally bounded in Ω. Let K be a compact subset in Ω, let x ∈ K and y ∈ Ω be such that |x − y| < δ ≤ 1

2dist(K, ∂Ω). By estimates on the Green

function we have ˆ

{y∈Ω:|x−y|<δ}

f(y)|x − y|2−ndy ≤ c ˆ

{y∈Ω:|x−y|<δ}

f(y)g(x, y)dy ≤ cuL∞(K) .

Giuseppe Di Fazio (University of Catania) 83 / 100

Lecture 4

PROOF. It means that the function ηK(δ) = sup

x∈K

ˆ

{y∈Ω:|x−y|<δ}

f(y) |x − y|n−2 dy is finite for some δ.

Giuseppe Di Fazio (University of Catania) 83 / 100

Lecture 4

CONDITION FOR CONTINUITY

THEOREM Let f ∈ L1(Ω), f ≥ 0 and u ∈ L1(Ω) be the very weak solution of the problem

• Lu = f

in Ω u = 0

• n ∂Ω.

Then, u ∈ C0(Ω) if and only if f ∈ Sloc(Ω).

Giuseppe Di Fazio (University of Catania) 84 / 100

Lecture 4

PROOF

PROOF. We already know that S implies continuity of solution. We show that if u ∈ C0(Ω) then f ∈ Sloc(Ω). We apply Dini Theorem about uniform convergence of sequences of continuous functions. If uδ(x) = ˆ

{y∈Ω:|x−y|≥δ}

f(y)g(x, y)dy and x0 ∈ Ω, by estimates for the Green function we have |f(y)||g(x, y)χBc

δ(x)(y) − g(x0, y)χBc δ(x0)| ≤

c δn−2 |f(y)| ∈ L1(Ω)

Giuseppe Di Fazio (University of Catania) 85 / 100

Lecture 4

PROOF

PROOF. By Lebesgue dominated convergence Theorem we have uδ(x) → uδ(x0) for δ > 0. Moreover, 0 ≤ uδ(x) ≤ u(x) and uδ → u everywhere in Ω. Then, for K compact subset of Ω, since u is continuous the convergence is uniform in K by Dini Theorem.

Giuseppe Di Fazio (University of Catania) 85 / 100

Lecture 4

PROOF

PROOF. Then sup

x∈K

(u(x) − uδ(x)) = sup

x∈K

ˆ

{y∈Ω:|x−y|<δ}

f(y)g(x, y)dy → 0 and, if 0 < δ < 1 2dist(K, ∂Ω), we have sup

x∈K

ˆ

{y∈Ω:|x−y|<δ}

f(y) |x − y|n−2 dy ≤ C sup

x∈K

ˆ

{y∈Ω:|x−y|<δ}

f(y)g(x, y)dy → 0.

Giuseppe Di Fazio (University of Catania) 85 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

Our next step is to show the necessary condition for the local H¨

• lder

continuity. The result will be achieved by a suitable Caccioppoli type inequality.

Giuseppe Di Fazio (University of Catania) 86 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

First we note that in this case the solution is the weak one. Indeed, it is locally bounded and then f ∈ ˜ Sloc(Ω) ⊂ H−1

loc (Ω).

Giuseppe Di Fazio (University of Catania) 87 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

For any Ω′ ⋐ Ω the function u is the weak solution of the equation Lu = f and u ∈ W 1,2

loc (Ω).

Indeed, if fk denote the truncation of f at level k we have 0 ≤ fk ≤ fk+1 ≤ f.

Giuseppe Di Fazio (University of Catania) 88 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

Denote by uk the weak solution of the Dirichlet problem

• Luk = fk

in Ω u = 0

• n ∂Ω

i.e. ˆ

Ω

A∇uk∇ϕ dx = ˆ

Ω

fkϕ dx for any ϕ ∈ W 1,2 (Ω).

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Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

Simply take ϕ = uk and get ∇uk2 ≤ νfk1uk∞ ≤ νf1u∞ . Since uk weakly converges in W 1,p (Ω) for 1 < p ≤

n n−1 then a

subsequence will converge in W 1,2 so it converges to u in W 1,2 .

Giuseppe Di Fazio (University of Catania) 90 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

THEOREM (CACCIOPPOLI TYPE INEQUALITY) Let f ∈ L1(Ω), f ≥ 0, and u ∈ L1(Ω) be the weak solution of the problem

• Lu = f

in Ω u = 0

• n ∂Ω

Then, for any ball Br ⊆ B2r ⋐ Ω, we have ˆ

Br

|∇u(x)|2 dx ≤ C

• r α

ˆ

B2r

f(x)dx + r n−2+2α

• .

Giuseppe Di Fazio (University of Catania) 91 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

PROOF. Let η be a cut-off function η(x) = 1 in Br, 0 ≤ η(x) ≤ 1, |∇η| ≤ c r . We can use ϕ ≡ η2(x)(u(x) − u2r) as a test function to obtain ˆ

Ω

A∇u∇ϕdx = ˆ

Ω

f(x)ϕ(x)dx for all ϕ ∈ W 1,2 (Ω).

Giuseppe Di Fazio (University of Catania) 92 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

PROOF. We have ˆ

Ω

A∇u

• 2η∇η(u − u2r) + η2∇u
• dx =

ˆ

Ω

f(x)η2(x)(u − u2r)dx and by ellipticity assumption ν−1 ˆ

B2r

|∇u(x)|2η2(x)dx ≤ 2 ˆ

B2r

aijuxiηxjη(x)|u(x) − u2r|dx + ˆ

B2r

f(x)η2(x)|u(x) − u2r|dx .

Giuseppe Di Fazio (University of Catania) 92 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

PROOF. Using the elementary inequality 0 ≤ 2ab ≤ εa2 + 1 ε b2 for all ε, a, b > 0 we have (ε = 1 4ν2 )

Giuseppe Di Fazio (University of Catania) 92 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

PROOF. Since 2 ˆ

B2r

A∇u∇ηη(x)|u − u2r|dx ≤ ν ˆ

B2r

2|∇u||∇η|η(x)|u − u2r|dx ≤ νε ˆ

B2r

|∇u|2η2(x)dx + ν 1 ε ˆ

B2r

|∇η|2|u − u2r|2dx = 1 4ν ˆ

B2r

|∇u|2η2(x)dx + 4ν3 ˆ

B2r

|∇η|2|u − u2r|2dx

Giuseppe Di Fazio (University of Catania) 92 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

PROOF. we have ν−1 ˆ

B2r

|∇u|2η2(x)dx ≤ 1 4ν ˆ

B2r

|∇u|2η2(x)dx + 4ν3 ˆ

B2r

|∇η|2|u(x) − u2r|2dx + ˆ

B2r

f(x)η2(x)|u(x) − u2r|dx

Giuseppe Di Fazio (University of Catania) 92 / 100

Lecture 4

CONDITION FOR H ¨

OLDER CONTINUITY

PROOF. that is, ˆ

B2r

|∇u|2η2(x)dx ≤ C ˆ

B2r

|∇η|2|u(x) − u2r|2dx + ˆ

B2r

f(x)η2(x)|u(x) − u2r|dx

• and by H¨
• lder continuity of the solution u we get the inequality.

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Lecture 4

STAMPACCHIA LEMMA

LEMMA Let ω : [0, R] → R be an increasing function. Let us assume that there exist 0 < η, α < 1 and H > 0 such that ω(r) ≤ ηω(4r) + H r α ∀r ≤ R . Then, there exist 0 < λ < 1 and c ≥ 0 such that ω(r) ≤ c r λ ∀r ≤ R .

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Lecture 4

PROOF OF STAMPACCHIA LEMMA

PROOF. Let a = η + 1 2 and β such that η4β = a < 1. We show that λ = min(α, β). Let M = sup

]R/4,R[

ω(̺) ̺λ and then ω(̺) ≤ M̺λ ∀̺ ∈]R/4, R[. Let n ∈ N be such that R 4n+1 ≤ ̺ < R 4n

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Lecture 4

PROOF OF STAMPACCHIA LEMMA

PROOF. By iteration we have ω(̺) ≤ ηω(4̺) + H̺α ≤ ηω(4̺) + H̺λ ≤ η

• ηω(42̺) + H(4̺)λ

+ H̺λ ≤ ηnM(4n̺)λ + H̺λ 1 1 − 4λη = ̺λ

• (4λη)nM +

H 1 − 4λη

• ≤ ̺λ
• M +

H 1 − a

• .

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Lecture 4

REMARK ON STAMPACCHIA LEMMA

REMARK Since β = log4 a η → ∞, if η → 0+ by suitable choose of β we may assume that λ = α.

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Lecture 4

H ¨

OLDER CONTINUITY

Now we are ready to show the necessary condition for H¨

• lder

continuity.

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Lecture 4

H ¨

OLDER CONTINUITY

THEOREM Let f ∈ L1(Ω), f ≥ 0, and let 0 < α < 1 be such that the weak solution u of the problem

• Lu = f

in Ω u = 0

• n ∂Ω

belongs to C0,α(Ω). Then f ∈ L1,n−2+α(Ω).

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Lecture 4

H ¨

OLDER CONTINUITY

PROOF. Let ¯ α the De Giorgi exponent for the operator L and let α be the exponent in our statement. If α ≥ ¯ α then there is nothing to prove.

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Lecture 4

H ¨

OLDER CONTINUITY

PROOF. We prove the result only in the case α < ¯ α. As in the previous result, the solution is the weak one because it is locally bounded. Indeed, if the solution is locally bounded, the function f must belong to ˜ S and then to W −1,2(Ω).

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Lecture 4

H ¨

OLDER CONTINUITY

PROOF. Now let ε be a positive number to be chosen later and let Br ⊆ B4r ⋐ Ω. If ϕ ∈ C∞

0 (B2r) 0 ≤ ϕ ≤ 1, ϕ ≡ 1 in Br, using the elementary inequality

2ab ≤ εr −αa2 + 1 ε r αb2 , we easily get

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Lecture 4

H ¨

OLDER CONTINUITY

PROOF. ˆ

Br

f(x)dx ≤ ˆ

B2r

f(x)ϕ(x)dx = ˆ

B2r

aij(x)uxiϕxjdx ≤ εCr −α ˆ

B2r

|∇u|2dx + 1 ε r α ˆ

B4r

|∇ϕ|2dx .

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Lecture 4

H ¨

OLDER CONTINUITY

PROOF. Now, by Caccioppoli inequality we obtain ˆ

Br

f(x)dx ≤ C

• ε

ˆ

B4r

f(x)dx + εr n−2+α + 1 ε r n−2+α

• that means

ˆ

Br

f(x)dx ≤ cε ˆ

B4r

f(x)dx + cεr n−2+α so by Stampacchia’s lemma there exists c such that ˆ

Br

f(x)dx ≤ cr n−2+α

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Lecture 4

SUMMARY OF LECTURES

1: Different concept of solution of a linear uniformly elliptic PDE 2: Green function for a linear uniformly elliptic operator and representation formula 3: Sufficient conditions for regularity 4: Necessary conditions for regularity - Motivation for the Stummel class and Morrey spaces.

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Lecture 4

THANKS Terima kasih atas perhatian anda

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