Higher integrability for doubly nonlinear parabolic equations Juha - - PowerPoint PPT Presentation

Higher integrability for doubly nonlinear parabolic equations

Juha Kinnunen, Aalto University, Finland juha.k.kinnunen@aalto.fi http://math.aalto.fi/∼jkkinnun/ August 26, 2019

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

References

• V. B¨
• gelein, F. Duzaar, J. Kinnunen and C. Scheven, Higher

integrability for doubly nonlinear parabolic systems, submitted (2018).

• V. B¨
• gelein, F. Duzaar, R. Korte and C. Scheven, The higher

integrability of weak solutions of porous medium systems,

• Adv. Nonlinear Anal. 8 (2018) 1004–1034.
• V. B¨
• gelein, F. Duzaar and C. Scheven, Higher integrability

for the singular porous medium system, submitted (2018).

• U. Gianazza and S. Schwarzacher, Self-improving property of

degenerate parabolic equations of porous medium-type, Amer.

• J. Math. (to appear).
• U. Gianazza and S. Schwarzacher, Self-improving property of

the fast diffusion equation, submitted (2018).

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Nonlinear parabolic equations

The porous medium equation/system ut − ∆(|u|m−1u) = 0, 0 < m < ∞. Sometimes written in the form (|u|m−2u)t − ∆u = 0, 1 < m < ∞. The parabolic p-Laplace equation/system ut − div(|Du|p−2Du) = 0, 1 < p < ∞. The doubly nonlinear equation/system (|u|p−2u)t − div(|Du|p−2Du) = 0, 1 < p < ∞. Sometimes called Trudinger’s equation. All equations above are special cases of a general doubly nonlinear equation/system (|u|m−2u)t − div(|Du|p−2Du) = 0, 1 < m < ∞, 1 < p < ∞.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Outline of the talk

Goal

To show that the gradient of a weak solution to (|u|p−2u)t − div(|Du|p−2Du) = 0, is locally integrable to a better power than assumed in the definition, with a reverse H¨

• lder inequality estimate for the

gradient.

Motivation

To extend the recent breakthroughs by Gianazza–Schwarzacher and others to cover a wider class of equations and systems. To develop direct methods that only apply energy estimates, Sobolev–Poincar´ e inequalities and Calder´

• n–Zygmund type

covering arguments. To develop methods that apply to sign-changing solutions and

• systems. In particular, Harnack estimates and the expansion of

positivity are not applied in the argument.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Review of the elliptic case

Let u ∈ W 1,p

loc (Ω), with 1 < p < ∞, be a weak solution of the

stationary p-Laplace equation div(|Du|p−2Du) = 0 in Ω ⊂ Rn. Then there exists ε > 0 such that

• B(x,r)

|Du|p+ε dy

• 1

p+ε

≤ c

B(x,2r)

|Du|p dy 1

p

for every ball B(x, 2r) ⊂ Ω. In particular, u ∈ W 1,p+ε

loc

(Ω). (Gehring 1973, Meyers and Elcrat 1975, Giaquinta and Modica 1979, Stredulinsky 1980)

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

A sketch of the proof

Step 1: An energy (Caccioppoli) estimate

• B(x,r)

|Du|p dy 1

p

≤ c r

• B(x,2r)

|u − uB(x,2r)|p dy 1

p

. Step 2: A Sobolev–Poincar´ e inequality

• B(x,2r)

|u − uB(x,2r)|p dy 1

p

≤ cr

• B(x,2r)

|Du|q dy 1

q

for some q < p.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Step 3: A reverse H¨

• lder inequality
• B(x,r)

|Du|p dy 1

p

≤ c

• B(x,2r)

|Du|q dy 1

q

, for every B(x, 2r) ⊂ Ω with some q < p. Step 4: The Gehring–Meyers–Elrcat lemma: There exists ε > 0 such that

• B(x,r)

|Du|p+ε dy

• 1

p+ε

≤ c

• B(x,2r)

|Du|p dy 1

p

for every B(x, 2r) ⊂ Ω.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Higher integrability results for parabolic equations 1(2)

Giaquinta–Struwe 1982: Parabolic systems with a quadratic structure, i.e. p = 2. Kinnunen–Lewis 2000: Systems of the parabolic p-Laplacian structure ut − div(|Du|p−2Du) = div(|F|p−2F), p >

2n n+2

Gianazza–Schwarzacher 2016: Nonnegative solutions to the porous medium type equations ut − ∆um = f , m ≥ 1.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Higher integrability results for parabolic equations 2(2)

B¨

• gelein–Duzaar–Korte–Scheven 2017: Systems of the

porous medium type ut − ∆

• |u|m−1u
• = div F,

m ≥ 1. Gianazza–Schwarzacher 2018: Nonnegative solutions to the porous medium type equations ut − ∆um = f ,

(n−2)+ n+2

< m < 1. B¨

• gelein–Duzaar–Scheven 2018: Systems of the porous

medium type ut − ∆

• |u|m−1u
• = div F,

(n−2)+ n+2

< m ≤ 1.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Potential applications

Partial regularity result for parabolic systems: Local H¨

• lder

continuity outside a small set (Misawa 2002). Nonlinear Calder´

• n-Zygmund theory (Acerbi and Mingione

2007). Estimates up to the boundary (Parviainen 2009, Moring-Scheven-Schwarzacher-Singer 2019). Higher order systems (Parviainen and B¨

• gelein 2010).

Stability of solutions as p varies (Kinnunen and Parviainen 2010).

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

The parabolic Sobolev space

Ω is an open subset of Rn and 0 ≤ t1 < t2 ≤ T. The space-time cylinders are ΩT = Ω × (0, T) and Dt1,t2 = D × (t1, t2), where D ⊂ Ω is an open set. The parabolic Sobolev space Lp(0, T; W 1,p(Ω)) consists of measurable functions u : ΩT → [−∞, ∞] such that x → u(x, t) belongs to W 1,p(Ω) for almost all t ∈ (0, T), and

• ΩT

(|u|p + |Du|p) dx dt < ∞. u ∈ Lp

loc(0, T; W 1,p loc (Ω)), if u belongs to the parabolic Sobolev

space for every Dt1,t2 ⋐ ΩT.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

The doubly nonlinear equation

Let 1 < p < ∞. A function u ∈ Lp

loc(0, T; W 1,p loc (Ω)) is a weak

solution to the doubly nonlinear equation (|u|p−2u)t − div(|Du|p−2Du) = 0 in ΩT, if

• ΩT
• |Du|p−2Du · Dϕ − |u|p−2uϕt
• dx dt = 0

for every ϕ ∈ C ∞

0 (ΩT).

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

The Barenblatt solution

Example The Barenblatt solution u(x, t) = t−

n p(p−1) exp

• − p−1

p

• |x|p

pt

• 1

p−1

, where x ∈ Rn and t > 0, is a solution to the doubly nonlinear equation in the upper half space. Observe: The Barenblatt solution is strictly positive for every x ∈ Rn and t > 0. This indicates that disturbances propagate with infinite speed.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

More general doubly nonlinear parabolic systems

We focus on the prototype equation, but it is possible to consider solutions u : ΩT → RN, N ≥ 1, to a system (|u|p−2u)t − div A(x, t, u, Du) = div(|F|p−2F) in ΩT where F : ΩT → RN and A: ΩT × RN × RNn → RNn with

• A(x, t, u, ξ) · ξ ≥ α|ξ|p ,

|A(x, t, u, ξ)| ≤ β|ξ|p−1, for almost every (x, t) ∈ ΩT and every u ∈ RN, ξ ∈ RNn with 0 < α ≤ β < ∞.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Structural properties

The equation is nonlinear: The sum of two solutions is not a solution, in general. Solutions can be scaled. Constants cannot be added to solutions. Thus the boundary values cannot be perturbed in a standard way by adding an epsilon. In the natural geometry a scaling by r in the spatial variable corresponds to rp in the time direction. For p = 2, this works for the heat equation.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

The Cole–Hopf transformation

Consider a nonnegative solution of (|u|p−2u)t − div(|Du|p−2Du) = 0. The transformation v = log u leads to the diffusive Hamilton–Jacobi equation vt −

1 p−1 div(|Dv|p−2Dv) = −|Dv|p,

with Dv = Du

u .

Note: In the elliptic case log u is a subsolution to the p-Laplace equation, but for the doubly nonlinear equation the equation changes.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

A change of variables

Consider a solution of (|u|p−2u)t − div(|Du|p−2Du) = 0. The transformation v = |u|p−2u leads to vt = div

• |Dv|

|v|

p−2 Dv

• .

Takeaway: The quotient |Du|

|u| appears again. This indicates that it

should play some role in the intrinsic geometry.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Some known results

Existence (Ivanov–Mkrtychyan–J¨ ager 1997, Sturm 2017) Asymptotic behavior (Manfredi-Vespri 1994, Savar´ e–Vespri 1993, Tedeev–Vespri 2015) Nonnegative weak solutions satisfy a scale and location invariant parabolic Harnack inequality in the space-time cylinders Qr,rp(x, t) = B(x, r) × (t − rp, t + rp). (Fornaro–Sosio–Vespri 2015, Gianazza–Vespri 2006, Kinnunen–Kuusi 2007, Trudinger 1968) Nonnegative weak solutions and are locally H¨

• lder continuous.

(Ivanov 1994, Kuusi–Lageoglu–Siljander–Urbano 2012, Porzio-Vespri 1993, Vespri 1992) The weak gradient of a positive weak solution is locally

• bounded. (Siljander 2010)

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Unexpected features

One might expect that the doubly nonlinear equation would have a similar behaviour as the heat equation, but this is not clear. The parabolic Harnack inequality does not immediately give local H¨

• lder continuity, because constants cannot be added to

solutions. The role of the intrinsic geometry is not clear. For example, the intrinsic geometry is not needed in Harnack estimates, but it is used in regularity theory. The question about uniqueness for the Dirichlet boundary value problem seems to be unsettled without additional

• assumptions. (Ivanov 1997, Lindgren-Lindqvist 2019)

Very little is known for sign-changing solutions. For example, continuity of a weak solution (and its gradient) seems to be an open question.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Higher integrability for the doubly nonlinear system

Theorem (B¨

• gelein, Duzaar, Kinnunen and Scheven 2018)

Let max 2n

n+2, 1

• < p <

2n (n−2)+

and assume that u is a weak solution to the doubly nonlinear equation in ΩT. There exists ε = ε(n, p) > 0 such that Du ∈ Lp(1+ε)

loc

(ΩT).

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

A Reverse H¨

• lder inequality for the gradient

The result comes with a uniform estimate

• Qr,rp

|Du|(1+ε)p dx dt ≤ c

• 1 +
• Q2r,(2r)p

|u|p (2r)p + |Du|p

• dx dt

ε

Q2r,(2r)p

|Du|p dx dt for every cylinder Q2r,(2r)p ⊂ ΩT. Notation: A ⊂ Rn+1, 0 < |A| < ∞,

• A

u dx dt = 1 |A|

• A

u(x, t) dx dt.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

The range of p

n ∈ {1, 2}: 1 < p < ∞. n ≥ 3:

2n n+2 < p < 2n n−2.

For the parabolic p-Laplace equation, the critical exponent is p >

2n n+2.

For the porous medium equation, the critical exponent is m > (n−2)+

n+2

and m ∼

1 p−1

⇐ ⇒ p <

2n (n−2)+ .

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Open questions

Does the result hold true in the entire range 1 < p < ∞ for n ≥ 3? Is the reverse H¨

• lder inequality true in the form
• Qr,rp

|Du|p+ε dx dt

• 1

p+ε

≤ c

• Q2r,(2r)p

|Du|p dx dt 1

p

for every cylinder Q2r,(2r)p ⊂ ΩT? Is the intrinsic geometry is really needed in the argument?

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

New features

The argument is self-contained, transparent and flexible. A new type of intrinsic geometry that depends both on u and Du is applied in the argument. Estimates for a power of a weak solution. A regularity result for the gradient. Result covers sign-changing solutions and doubly nonlinear systems. Proof can be extended to more general doubly nonlinear systems.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Strategy of the proof

Step 1: An energy estimate on intrinsic cylinders. Step 2: A Sobolev inequality on intrinsic cylinders. Step 3: A reverse H¨

• lder inequality for the gradient on

intrinsic cylinders. Step 4: A Calder´

• n–Zygmund type stopping time argument

to construct a collection of intrinsic cylinders {Qr,s} with a Vitali covering property. Step 5: A covering of the distribution set {z : |Du|(z) > λ} by intrinsic cylinders Qr,s as constructed above satisfying λp ≈

• Qr,s

|Du|p dx dt. Step 6: Cavalieri’s principle gives an estimate for

• Qr,rp

|Du|(1+ε)p dx dt.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Intrinsic geometry 1(3)

We discuss a formal motivation for an appropriate intrinsic geometry. Let Qr,s = Qr,s(z0) = B(x0, r) × (t0 − s, t0 + s) ⊂ ΩT with z0 = (x0, t0). Consider v(y, t) = u(ry, sτ) for (y, t) ∈ B(0, 1) × (−1, 1). Infinitesimally |u| ≈

• Qr,s

|u|p dx dt 1

p

and |Du| ≈

• Qr,s

|Du|p dx dt 1

p

.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Intrinsic geometry 2(3)

∂τv = s∂tu ≈ s(|u|p−2u)t

• Qr,s

|u|p dx dt p−2

p

= s div(|Du|p−2Du)

Qr,s

|u|p dx dt p−2

p

≈ s

• Qr,s

|Du|p dx dt p−2

p

Qr,s

|u|p dx dt p−2

p

∆xu = s

• Qr,s

|Du|p dx dt p−2

p

r2

• Qr,s

|u|p dx dt p−2

p

∆yv.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Intrinsic geometry 3(3)

We choose the scaling so that the factor in front of ∆yv is one so that the equation looks like the heat equation in its own geometry. This gives s = r2

• Qr,s

|u|p dx dt p−2

p

• Qr,s

|Du|p dx dt p−2

p

= rp

• Qr,s

|u|p rp dx dt p−2

p

• Qr,s

|Du|p dx dt p−2

p

and thus s rp = µp−2 with µp =

• Qr,s

|u|p rp dx dt

• Qr,s

|Du|p dx dt . Takeaway: This indicates homogeneous behavior on intrinsic cylinders of the form Qr,s with

s rp = µp−2.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

An energy estimate

Notation: uα = |u|α−1u, α > 0 (0α = 0). For every a ∈ R, we have sup

t∈(t0−s,t0+s)

• B(x0,r)

|u(x, t)

p 2 − a p 2 |2

S dx +

• Qr,s(z0)

|Du|p dx dt ≤ c

• QR,S(z0)

|u

p 2 − a p 2 |2

S − s + |u − a|p (R − r)p

• dx dt,

where QR,S(z0) = B(x0, R) × (t0 − S, t0 + S) ⊂ ΩT with R, S > 0, and r ∈ R

2 , R

• , s ∈

S

2p , S

• .

Takeaway: The energy estimate is for a power of a solution. The power appears in the boundary term in the proof of the energy estimate.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

An energy estimate on intrinsic cylinders

By choosing S = µp−2Rp and s = µp−2rp, we have sup

t∈(t0−s,t0+s)

• B(x0,r)

µ2−p|u(x, t)

p 2 − a p 2 |2

Rp dx +

• Qr,s(z0)

|Du|p dx dt ≤ c

• QR,S(z0)

µ2−p|u

p 2 − a p 2 |2

Rp − rp + |u − a|p (R − r)p

• dx dt

with µp ≈

• Qr,s(z0)

|u|p rp dx dt

• Qr,s(z0)

|Du|p dx dt . Takeaway: The energy estimate becomes homogeneous in the intrinsic geometry.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

A gluing lemma

Let QR,S(z0) = B(x0, R) × (t0 − S, t0 + S) ⊂ ΩT with R, S > 0. There exists r ∈ R

2 , R

• such that
• B(x0,r)

u(x, t2)p−1 dx −

• B(x0,r)

u(x, t1)p−1 dx

• ≤ c S

R

• QR,S(z0)

|Du|p−1 dx dt for every t1, t2 ∈ (t0 − S, t0 + S). Takeaway: The gluing lemma enables us to pass from slice-wise integral averages in Sobolev–Poincar´ e inequalities to space-time integral averages in energy estimates. It compensates the lack of differentiability with respect to time.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

A Sobolev-Poincar´ e inequality

Let p >

2n n+2. There exists q < p such that

• Qr,s(z0)

µ2−p|u

p 2 − a p 2 |2

rp + |u − a|p rp

• dx dt

≤ c

• Qr,s(z0)

|Du|q dx dt p

q

+ other terms for every intrinsic cylinder Qr,s(z0). Takeaway: This version of the Sobolev-Poincar´ e inequality holds for weak solutions of the doubly nonlinear equation. The gluing lemma is used in the argument.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

A reverse H¨

• lder inequality

A combination of the energy estimate and the Sobolev-Poincar´ e inequality gives a reverse H¨

• lder inequality. Let p >

2n n+2. Then

there exists q < p such that

• Qr,s(z0)

|Du|p dx dt ≤ c

• Qr,s(z0)

|Du|q dx dt p

q

for every intrinsic cylinder Qr,s(z0). Takeaway: The reverse H¨

• lder inequality holds in the intrinsic

geometry.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

The proof completed

The Calder´

• n–Zygmund type stopping time argument to

construct a collection of cylinders {Qr,s(z)} with a Vitali covering property and a covering of the distribution set {z : |Du|(z) > λ} by intrinsic cylinders Qr,s(z) is a modification of the argument by Gianazza-Schwarzacher. The fact that Cavalieri’s principle gives an estimate for

• Qr,rp

|Du|(1+ε)p dx dt is rather standard.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations

Extensions and possible future developments

Higher integrability up to the boundary. General doubly nonlinear systems with different powers. Stability of solutions with respect to p. Very weak solutions.

Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations