Suitable Solution Concept for a Nonlinear Elliptic PDE Noureddine - - PowerPoint PPT Presentation
Suitable Solution Concept for a Nonlinear Elliptic PDE Noureddine Igbida Institut de recherche XLIM, UMR-CNRS 6172 Universit e de Limoges 87060 Limoges, France Colloque EDP-Normandie 2011 Rouen, 25-26 Octobre 2011 N. Igbida Suitable
Noureddine Igbida
Institut de recherche XLIM, UMR-CNRS 6172 Universit´ e de Limoges 87060 Limoges, France
Colloque EDP-Normandie 2011 Rouen, 25-26 Octobre 2011
Suitable Solution Concept for a Nonlinear Elliptic PDE
Collaboration
Soma Safimba and Stanislas Ouaro : University of Ouaga-Dougou (Burkina Faso) Fahd Karami : University of Essaouira (Maroc) β(u) − ∇ · a(x, ∇u) ∋ µ in Ω γ(u) + a(x, ∇u) · η ∋ ψ
Ω ⊂ RN is a bounded regular domain ∂Ω =: Γ is the boundary γ and β are maximal montone graphes in R × R a is a Leray-Lions type operator ; i.e. a : Ω × RN → RN is Carath´ eodory and satisfies
H1 : a(x, ξ) · ξ λ|ξ|p, λ > 0, 1 < p < ∞. H2 : |a(x, ξ)| ≤ σ(g(x) + |ξ|p−1), g ∈ Lp
′
(Ω), σ > 0. H3 : (a(x, ξ) − a(x, η)) · (ξ − η) > 0.
Questions Existence and uniqueness of the solution in the case where µ ∈ Lp′(Ω), ψ ∈ Lp′(Γ). µ ∈ L1(Ω), ψ ∈ L1(Γ). µ and ψ are Radon measures (diffuse).
Suitable Solution Concept for a Nonlinear Elliptic PDE
The case D(β) = R. µ ∈ Mb(Ω) is diffuse β(u) − ∇ · a(x, ∇u) ∋ µ in Ω u = 0
ψ, µ ∈ Lp′ β(u) − ∇ · a(x, ∇u) ∋ µ in Ω γ(u) + a(x, ∇u) · η ∋ ψ
Suitable Solution Concept for a Nonlinear Elliptic PDE
I - Examples, definitions, reminders II - Examples of nonexistence and our main results III - Main ideas of the proofs
Suitable Solution Concept for a Nonlinear Elliptic PDE
Suitable Solution Concept for a Nonlinear Elliptic PDE
Nonlinear and linear diffusion : a(x, ∇u) = ∇u : Laplace operator a(x, ∇u) = |∇u|p−1∇u : p−Laplace operator, 1 < p < ∞ β(r) = |r|αr : porous medium equation β(r) = (r − 1)+ − (r − 1)− : Stefan problem β = H (Heaviside graph) : Hele-Shaw problem β = H−1 : Obstacle problem Boundary condition : γ ≡ 0 : Nonhomogeneous Neumann boundary condition D(γ) = {0} : Dirichlet boundary condition Applications : Heat equation, nonlinear diffusion in porous medium Stefan problem, Hele-Shaw problem, Obstacle problem .....
Suitable Solution Concept for a Nonlinear Elliptic PDE
Weak solution : is a couple (u, z) ∈ W 1,p (Ω) × L1(Ω) such that z ∈ β(u), LN − a.e. in Ω, and −∇ · a(x, ∇u) = µ − z, in D′(Ω). In general, if the data are enough regular, weak solutions does not exist. Renormalized solution : is a couple of measurable function (u, z) such that z ∈ L1(Ω), Tku ∈ W 1,p (Ω), for any k 0, z ∈ β(u) a.e. in Ω and −∇ · h(u)a(x, Du) + h(u)z = h(u)µ − h′(u) a(x, ∇u) · ∇u, in D′(Ω), for any h ∈ Cc(R), and lim
n→∞
|∇u|p dx = 0. Entropic solution : is a couple of measurable function (u, z) such that z ∈ L1(Ω), Tku ∈ W 1,p (Ω), for any k 0, z ∈ β(u) a.e. in Ω and
a(x, ∇u) · ∇Tk(u − ξ) +
z Tk(u − ξ) ≤
µ Tk(u − ξ), for any k 0 and ξ ∈ W 1,p (Ω) ∩ L∞(Ω).
Suitable Solution Concept for a Nonlinear Elliptic PDE
In the case of Neumann boundary condition one needs to take the test function ξ ∈ W1,p(Ω). And, for renormalized/entropic solution, one needs to work in T 1,p
tr (Ω).
In many cases, an enropic/renormalized solution is a solution in the sense of distribution : (u, z) ∈ W 1,1 (Ω) × L1(Ω) such that z ∈ β(u), LN − a.e. in Ω, and −∇ · a(x, ∇u) = µ − z, in D′(Ω). In the case where a(x, ξ) = ξ, there is an equivalence between the entropic solution and the solution in the sense of distribution. In general, a solution in the sense of distribution is not unique. In the case where a(x, ξ) = ξ, a solution in the sense of distribution is unique. A weak solution is an enropic/renormalized solution. If (u, w) is an enropic/renormalized solution and u ∈ L∞(Ω), then (u, z) is a weak solution.
Suitable Solution Concept for a Nonlinear Elliptic PDE
Dirichlet boundary condition β(u) − ∇ · a(x, ∇u) ∋ µ in Ω u = 0
µ ∈ W −1,p′(Ω) : (P) is well posed in the sense of weak solution. µ ∈ L1(Ω) : (P) is well posed in the sense of renormalized solution. µ ∈ Mb(Ω) : (P) is well posed in the sense of renormalized solution if
D(β) = R µ is diffusive.
Recall that, a Radon measure µ is said to be diffuse with respect to the capacity W 1,p (Ω) (p−capacity for short) if µ(E) = 0 for every set E such that capp(E, Ω) = 0. The p−capacity of every subset E with respect to Ω is defined as : capp(E, Ω) = inf
Ω
|∇u|pdx ; u ∈ W 1,p (Ω), u 0, s.t. u = 1 a.e. E
The set of diffuse measures is denoted by Mp
b(Ω).
Suitable Solution Concept for a Nonlinear Elliptic PDE
General boundary condition β(u) − ∇ · a(x, ∇u) ∋ µ in Ω γ(u) + a(x, ∇u) · η ∋ ψ
µ, ψ ∈ Lp′ : (P) is well posed in the sense of weak solution when
(A) D(β) = R (B) ψ ≡ 0
µ, ψ ∈ L1 : (P) is well posed in the sense of of renormalized solution when
(A) D(β) = R (B) ψ ≡ 0
µ ∈ Mb(Ω) : (P) is well posed in the sense of renormalized solution if
D(β) = R µ is diffusive.
Remark The non existence of standard solution appears in the following cases : D(β) = R Dirichlet boundary condition and Radon measure µ (diffuse) D(β) = R Non homogeneous Neumann boundary condition.
Suitable Solution Concept for a Nonlinear Elliptic PDE
Very large litterature : Dirichlet and homogeneous Neumann boundary condition : B´ enilan, Brezis, Boccardo, T Gallouet, Murat, Blanchard, Crandall, Redouan, Guib´ e, Porreta, Dal Maso, Orsina, Prignet, NI, Mazon, Toledo, Andreu ..... Nonhomogeneous Neumann boundary condition
Laplacien with β ≡ 0 and γ continuous : J. Hulshof, 1987 Laplacien with γ and β continuous in R : N. Kenmochi, 1990 Laplacien with γ ≡ 0 and β continuous (not everywhere defined) : NI, 2002/06. General cases : F. Andreu, J. Mazon, J. Toledo, NI : 2008-2011
Suitable Solution Concept for a Nonlinear Elliptic PDE
In the particular case a(x, ξ) = ξ, the problem reads −∆u + β(u) ∋ µ in Ω ∂ηu + γ(u) ∋ ψ
IdR , γ a maximal monotone graph and µ, ψ ∈ L2 Brezis-Strauss : µ ∈ L1(Ω), ψ ≡ 0 and γ, β continuous nondecreasing functions from R into R with β′ ǫ > 0.
enilan, M. G. Crandall and P. Sacks : γ and β maximal monotone graphs in R2 such that 0 ∈ γ(0) ∩ β(0), µ ∈ L1 and ψ ≡ 0.
Suitable Solution Concept for a Nonlinear Elliptic PDE
For the general case
dans Ω γ(u) − a(x, ∇u) · η ∋ ψ sur ∂Ω 90ies ”truncation and renormalization” allowed to characterize the solutions intrinsically, in a way acceptable for the PDE community and became classical in a few years Boccardo and Gallouet, 1992 : Dirichlet boundary condition, β ≡ 0 and µ a Radon measure. Murat, 1994 : Dirichlet boundary condition, µ a Radon measure. B´ enilan, Boccardo, Gallouet, Gariepy, Pierre and Vazquez, 1995 : Dirichlet boundary condition and µ ∈ L1 Dal Maso, Murat, Orsina and Prignet, 1999 : Dirichlet boundary condition and µ a Radon measure. Blanchard and Murat, 1997 : ”parabolic case”.
entropy/renormalized) solutions in the case
µ and ψ ∈ L1(∂Ω), in the cases
(A) D(β) = R (B) ψ ≡ 0
µ and ψ are two Radon diffuse measures, in the case
(A) D(β) = R
Suitable Solution Concept for a Nonlinear Elliptic PDE
Suitable Solution Concept for a Nonlinear Elliptic PDE
Our aim here is to treat the case where D(β) = R. Examples Signorini problem (elasticity), unilateral constraint : β(r) = ∅ if r < 0 ] − ∞, 0] if r = 0 if r > 0, Optimal control problem, modeling of semipermeability : β(r) = ∅ if r < m ] − ∞, 0] if r = m if r ∈]m, M[ [0, +∞[ if r = M ∅ if r > M, where m < 0 < M.
Suitable Solution Concept for a Nonlinear Elliptic PDE
Dirichlet boundary condition Example 1 : β be the maximal monotone graph given by β(r) = if r < 0 [0, ∞) if r = 0. If µ is nonnegative and (u, z) is a solution of (P), then 0 ≤
z Tku +
a(x, ∇u) · ∇Tku =
Tkuµ ≤ 0, for any k 0. This implies that u ≡ 0 and z = µ ⇒ if µ is a nonnegative Radon measure. Example 2 : β : R → R continuous nondecreasing, β(0) = 0 and lim
t↑1 β(t) = +∞
and lim
r→1−(1 − t)
2−λ λ g(t) > 0
µ with respect to the capacity H1(Ω) such that the problem −∆u + β(u) = µ in Ω, u = 0
has no weak solution. That measure µ is taken such µ+ < < HN−2+α, for some 0 < α < 2, where Hs denotes the s−dimensional Hausdorff measure.
Suitable Solution Concept for a Nonlinear Elliptic PDE
Non homogeneous Neumann boundary condition −∆u + β(u) ∋ µ in Ω ∂ηu = ψ
β is a maximal monotone graph, D(β) = [0, 1] and 0 ∈ β(0) µ ≤ 0 a.e. in Ω ψ ≤ 0 a.e. in ∂Ω. If (u, w) is a solution then, 0 ≤ u ≤ 1 a.e. in Ω 0 ≤
|Du|2 +
zu =
µu +
ψu ≤ 0 ⇓ ⇓ ⇓ ⇓ u is constant and
zv =
µv +
ψv, ∀ v ∈ H1(Ω) ∩ L∞(Ω) ⇓ ⇓ ⇓ ⇓ µ = z a.e. in Ω and ψ ≡ 0 a.e. in ∂Ω. ⇓ ⇓ ⇓ ⇓ Nonexistence if 0 ψ ≡ 0 and µ ≤ 0 (even for regular µ and ψ)
Suitable Solution Concept for a Nonlinear Elliptic PDE
⇓ ⇓ ⇓ ⇓ β(u) − ∆u ∋ µ in Ω u = 0
A ”good measure” is a Radon measure µ such that (P) has a weak solution. The ”reduced measure” denoted by µ∗ associated with µ is the right measure that we can associate with µ such that (P) with µ replaced by µ∗ has a solution. Use a natural approximation scheme (keep µ fixed and approximate β or keep β fixed and approximate µ), pass to the limit in the equation and characterize the right part of µ for which the problem is well posed. If D(β) = R, then any diffuse Radon measure is a good measure. There exists β, such that D(β) = 0, for which any diffuse Radon measure is necessary a good measure (L. Dupaigne, A. Ponce and A. Porretta).
Suitable Solution Concept for a Nonlinear Elliptic PDE
D(β) = [m, M]. Theorem (NI, S. Safimba and S. Oauro,JDE-2011) For any µ ∈ Mp
b(Ω), the problem
β(u) − ∇ · a(x, ∇u) ∋ µ in Ω u = 0
has a unique solution (u, z) in the sense that (u, z) ∈ W 1,p (Ω) × L1(Ω), z ∈ β(u) LN − a.e. in Ω there exists ν ∈ Mp
b(Ω) such that ν ⊥ LN
ν+ is concentred on [u = M] ν− is concentred on [u = m] for any ξ ∈ W 1,p (Ω) ∩ L∞(Ω)
a(x, ∇u) · ∇ξdx +
zξdx +
ξdν =
ξdµ. Moreover, we have ν+ < < µs⌊ [u = M] and ν− < < µs⌊ [u = m].
Suitable Solution Concept for a Nonlinear Elliptic PDE
D(β) = [m, M]. Theorem (F. Andreu, NI, J. Mazon and J. Toledo,M3AS-2008) For any µ ∈ Lp′(Ω) and ψ ∈ Lp′(∂Ω), the problem β(u) − ∇ · a(x, ∇u) ∋ µ in Ω a(x, ∇u) · η = ψ
has a unique solution (u, z) in the sense that (u, z) ∈ W 1,p(Ω) × L1(Ω), z ∈ β(u) LN − a.e. in Ω there exists ν ∈ L1(∂Ω) such that ν+ is concentred on [u = M] ν− is concentred on [u = m] for any ξ ∈ W 1,p (Ω) ∩ L∞(Ω)
a(x, ∇u) · ∇ξdx +
zξdx +
ξν dx =
ξdµ.
Suitable Solution Concept for a Nonlinear Elliptic PDE
Assume that (u, z) is the solution in the sense of : (u, z) ∈ W 1,p (Ω) × L1(Ω), z ∈ β(u) LN − a.e. in Ω there exists ν ∈ Mp
b(Ω) such that ν ⊥ LN
ν+ is concentred on [u = M] ν− is concentred on [u = m] and
a(x, ∇u) · ∇ξdx +
zξdx +
ξdν =
ξdµ, ∀ξ ∈ W 1,p (Ω) ∩ L∞(Ω). (u, z) is also the unique solution in the following sense :
(u, z) ∈ W 1,p (Ω) × L1(Ω), z ∈ β(u) LN − a.e. in Ω for any ξ ∈ W 1,p (Ω) s.t. m ≤ ξ ≤ M,
a(x, ∇u) · ∇(u − ξ) +
z (u − ξ) ≤
µ (u − ξ).
For the Laplace operator β(u) − ∆u ∋ µ in Ω u = 0
the reduced measure is given by µ∗ = µ − ν [u = m] − ν [u = M]
Suitable Solution Concept for a Nonlinear Elliptic PDE
Suitable Solution Concept for a Nonlinear Elliptic PDE
Take an approximation : β ← βn au sens des graphes βn(un) − ∇ · a(x, ∇un) = µ in Ω Aims : S1 : un converges weakly in W 1,p(Ω) S2 : zn := βn(un) converges in L1(Ω) (at least weakly) S3 : Identification :
identification of lim
n→∞ a(x, ∇un) : Minty type arguments
identification of lim
n→∞ βn(un) : monotony, strong/weak convergence of un/βn(un)
Dirichlet boundary condition S1 : Test with un (+) Poincar´ e inequality = ⇒ un is bounded in W 1,p(Ω) S2 : Estimates in Lq, for any q 1, = ⇒ zn converges in L1(Ω)-weakly See here that there is no restriction on D(β) ⇓ ⇓ ⇓ ⇓ Existence of weak solution for any maximal monotone graph β and µ ∈ Lp′(Ω)
Suitable Solution Concept for a Nonlinear Elliptic PDE
Take an approximation : β ← βn au sens des graphes βn(un) − ∇ · a(x, ∇un) = µ in Ω Aims : S1 : un converges weakly in W 1,p(Ω) S2 : zn := βn(un) converges in L1(Ω) (at least weakly) S3 : Identification :
identification of lim
n→∞ a(x, ∇un) : Minty type arguments
identification of lim
n→∞ βn(un) : monotony, strong/weak convergence of un/βn(un)
Neumann boundary condition S1 : un − 1 |Ω|
un is bounded in W 1,p(Ω) Additional assumptions (necessary condition) = ⇒ 1 |Ω|
un is bounded S2 : Monotone approximation for D(β) = R : βm
n := β + 1
m I + − 1 n I − = ⇒ zm
n converges in L1(Ω).
See here that D(βm
n ) = D(β)
⇓ ⇓ ⇓ ⇓ Existence of weak solution for any maximal monotone graph such that D(β) = R and µ, ψ ∈ Lp′
Suitable Solution Concept for a Nonlinear Elliptic PDE
Assume we have existence of a weak solution for regular data µ. Take an approximation : µ ← µn in L1(Ω) β(un) − ∇ · a(x, ∇un) = µn in Ω Main Steps : S1 : The convergence of un : the estimates in W 1,p(Ω) falls to be true !!
Tk(un) ⇀ Tk(u) weakly in W 1,p(Ω) a(x, ∇Tk(un)) ⇀ a(x, ∇Tk(u)) weakly in Lp′(Ω) ∇Tk(un) − → ∇Tk(u) a.e in Ω a(x, ∇Tk(un)) · ∇Tk(un) − → a(x, ∇Tk(u)) · ∇Tk(u) a.e in Ω and strongly in L1(Ω) ∇Tk(un) − → ∇Tk(u) strongly in Lp(Ω)
S2 : The convergence of zn The contraction property = ⇒ zn converges in L1(Ω) ⇓ ⇓ ⇓ ⇓ Existence of a renormalized solution :
Suitable Solution Concept for a Nonlinear Elliptic PDE
µ is diffuse = ⇒ µ = f + ∇ · F, f ∈ L1(Ω), F ∈ Lp′(Ω)N Take an approximation βε(uε) − ∇ · a(x, ∇uε) = fε + ∇ · Fε in Ω Main Steps : S1 : The convergence of uǫ :
Tk(uǫ) ⇀ Tk(u) weakly in W 1,p(Ω) a(x, ∇Tk(uǫ)) ⇀ a(x, ∇Tk(u)) weakly in Lp′(Ω) ∇Tk(uǫ) − → ∇Tk(u) a.e in Ω a(x, ∇Tk(uǫ)) · ∇Tk(uǫ) − → a(x, ∇Tk(u)) · ∇Tk(u) a.e in Ω and strongly in L1(Ω) ∇Tk(uǫ) − → ∇Tk(u) strongly in Lp(Ω)
S2 : The convergence of zǫ := βǫ(uǫ)
zǫ is bounded in L1(Ω) zǫ → z in Mb(Ω) − weak∗ Radon-Nykodym decomposition relatively to Lebesgue measure : z = zr + zs
The characterization of zr and zs ???
Suitable Solution Concept for a Nonlinear Elliptic PDE
Aims u(x) ∈ D(β), LN−a.e. x ∈ Ω zr(x) ∈ β(u(x)), LN−a.e. x ∈ Ω z+
s
is concentred on [u = M] and z−
s
is concentred on [u = m] ⇑ ⇑ ⇑ ⇑ We have (here) A sequences (uǫ)ǫ>0 and (zǫ)ǫ>0 of measurable functions on Ω satisfying zǫ = βǫ(uǫ) LN − a.e in Ω uǫ − → u in L1(Ω) zǫ → z = zr + zs in Mb(Ω) − weak∗ βε → β in the sense of graphe
Suitable Solution Concept for a Nonlinear Elliptic PDE
u(x) ∈ D(β), LN−a.e. x ∈ Ω ? Lemma (NI, S. Oauro, S. Safimba, 2011) Let (βn)n1 be a sequence of maximal monotone graphs such that βn − → β in the sense of graphs. We consider (zn)n1 and (un)n1 two sequences of L1(Ω), such that zn ∈ βn(un), LN−a.e. in Ω, for any n = 1, 2, ... If (zn)n1 is bounded in L1(Ω) and un − → u in L1(Ω), then u ∈ dom(β) LN − a.e. in Ω. The main tool for the proof of this Lemma is the ”bitting lemma of Chacon” Lemma The “bitting lemma of Chacon” Let Ω ⊂ RN be an open bounded of RN and (fn)n a bounded sequence in L1(Ω). Then there exist f ∈ L1(Ω), a subsequence (fnk )k and a sequence of measurable sets (Ej)j, Ej ⊂ Ω, ∀j ∈ N with Ej+1 ⊂ Ej and lim
j→+∞ |Ej| = 0, such that for any j ∈ N, fnk ⇀ f in L1
Ω\Ej
Suitable Solution Concept for a Nonlinear Elliptic PDE
(zn)n1 is bounded in L1(Ω) “bitting lemma of Chacon” ⇒ there exist
z ∈ L1(Ω), a subsequence (znk )k1 a sequence of mesurable sets (Ej)j∈N in Ω such that Ej+1 ⊂ Ej, ∀j ∈ N, lim
j→+∞ |Ej| = 0
and ∀j ∈ N, znk ⇀ z in L1(Ω\Ej).
unk − → u, in L1(Ω) (and in L1(Ω\Ej), ∀ j ∈ N) βnk − → β in the sense of graphs ⇓ ⇓ ⇓ ⇓ z ∈ β(u), a.e in Ω\Ej ⇓ ⇓ ⇓ ⇓ u ∈ dom(β), a.e in Ω\Ej, ∀ j ∈ N ⇓ ⇓ ⇓ ⇓ u ∈ dom(β), a.e in Ω.
Suitable Solution Concept for a Nonlinear Elliptic PDE
We have A sequences (uǫ)ǫ>0 and (zǫ := βǫ(uǫ))ǫ>0 of measurable functions uǫ − → u, LN − a.e in Ω, zn → z = zr + zs, in Mb(Ω) − weak∗ βε → β, in the sense of graphe Assume here, that z = µ + ∇ · a(x, ∇u) in D′(Ω) (+) βε = ∂jǫ, jε is lower semi-continuous functions and jǫ(r) 0, ∀ǫ > 0 and jǫ ↑ j as ǫ ↓ 0. u ∈ dom(j), LN − a.e in Ω, Lemma Assume moreover, that dom(j) = [m1, m2] and z ∈ Mp
b(Ω) satisfies
lim inf
ǫ→0
(t − uǫ) ξ zǫdx
(t − u)ξdz, ∀t ∈ R, (1) ∀ξ ∈ C 1
c (Ω), ξ 0, Then
z = zrLN + zs with zs⊥LN, zr ∈ ∂j(u) LN − a.e in Ω, zr ∈ L1(Ω), z+
s
is concentrated on [u = m2] and z−
s
is concentrated on [u = m1].
Suitable Solution Concept for a Nonlinear Elliptic PDE
j(t) jǫ(t) jǫ(uǫ) + (t − uǫ)zǫ, LN − a.e in Ω
ξj(t)dx
ξj˜
ǫ(uǫ)dx +
(t − uǫ)ξzǫdx. Fatou’s Lemma implies that
ξj(t)dx
ξj˜
ǫ(u)dx + lim inf ǫ→0
(t − uǫ)ξzǫdx. The assumption of the lemma implies that
ξj(t)dx
ξj˜
ǫ(u)dx +
ξ(t − u)dz. Letting ˜ ǫ → 0, and using again Fatou’s Lemma, we get
ξj(t)dx
ξj(u)dx +
ξ(t − u)dz Then j(t) j(u) + (t − u)z, in Mb(Ω), ∀t ∈ R Comparing the regular part and the singular part, we obtain j(t) j(u) + (t − u)zr, LN − a.e , ∀t ∈ R ⇒ zr(x) ∈ β(u(x)), LN − a.e. and (t − u)zs ≤ 0 in Mb(Ω), ∀t ∈ dom(j) ⇒ zs([m < u < M]) = 0.
Suitable Solution Concept for a Nonlinear Elliptic PDE
Lemma (NI-SO-SS,2011 and FA-NI-JM-JT,2008) Let η ∈ W 1,p (Ω), κ ∈ Mb(Ω) and λ ∈ R be such that η ≤ λ a.e in Ω (resp. η λ), giry@crous − limoges.fr = −div a(x, ∇η) in D′(Ω). Then
ξdκ 0 (resp.)
ξdκ ≤ 0, for any ξ ∈ C 1
c (Ω), ξ 0.
−∇ · a(x, ∇u) = µ − z =: κ in D′(Ω) m ≤ u ≤ M ⇓ ⇓ ⇓ ⇓ z+
s <
< µs⌊ [u = M] and z−
s
< < µs⌊ [u = m].
Suitable Solution Concept for a Nonlinear Elliptic PDE
Suitable Solution Concept for a Nonlinear Elliptic PDE
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Suitable Solution Concept for a Nonlinear Elliptic PDE