M. Warma
Universidad de Puerto Rico, Rio Piedras, Mathematics, Faculty Member
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ABSTRACT Let Omega subset of R-N (N >= 2) be a bounded domain with a boundary partial derivative Omega of class C-2 and let alpha, beta be maximal monotone graphs in R-2 satisfying alpha(0) boolean AND beta (0) there exists 0.... more
ABSTRACT Let Omega subset of R-N (N >= 2) be a bounded domain with a boundary partial derivative Omega of class C-2 and let alpha, beta be maximal monotone graphs in R-2 satisfying alpha(0) boolean AND beta (0) there exists 0. Given f is an element of L-1(Omega) and g is an element of L-1(partial derivative Omega), we characterize the existence and uniqueness of weak solutions to the semi-linear elliptic equation -Delta u+alpha(u) there exists f in Omega with the nonlinear general Wentzell boundary conditions -Delta(Gamma)u+partial derivative u/partial derivative v + beta(u)) there exists g on partial derivative Omega. We also show the well-posedness of the associated parabolic problem on the Banach space L1(Omega) x L-1(partial derivative Omega).
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Let p∈(1, N), Ω⊂ RN a bounded W1, p-extension domain and let μ be an upper d-Ahlfors measure on∂ Ω with d∈(N− p, N). We show in the first part that for every p∈[2N/(N+ 2), N)∩(1, N), a realization of the p-Laplace operator with... more
Let p∈(1, N), Ω⊂ RN a bounded W1, p-extension domain and let μ be an upper d-Ahlfors measure on∂ Ω with d∈(N− p, N). We show in the first part that for every p∈[2N/(N+ 2), N)∩(1, N), a realization of the p-Laplace operator with (nonlinear) generalized nonlocal ...
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We characterize the Lq-solvability of a class of quasi-linear elliptic equations involving the p-Laplace operator with generalized nonlinear Robin type boundary conditions on bad domains. Some uniqueness results are also given.
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ABSTRACT We show that on a bounded domain Ω⊂ℝ N with Lipschitz continuous boundary ∂Ω, weak solutions of the elliptic equation λu-Au=f in Ω with the boundary conditions -γΔ Γ u+∂ ν a u+βu=g on ∂Ω are globally Hölder continuous on Ω ¯.... more
ABSTRACT We show that on a bounded domain Ω⊂ℝ N with Lipschitz continuous boundary ∂Ω, weak solutions of the elliptic equation λu-Au=f in Ω with the boundary conditions -γΔ Γ u+∂ ν a u+βu=g on ∂Ω are globally Hölder continuous on Ω ¯. Here A is a uniformly elliptic operator in divergence form with bounded measurable coefficients, Δ Γ is the Laplace-Beltrami operator on ∂Ω, ∂ ν a u denotes the conormal derivative of u, λ,γ>0 are real numbers and β is a bounded measurable function on ∂Ω. We also obtain that a realization of the operator A in C(Ω ¯) with the general Wentzell boundary conditions (Au)| ∂Ω -γΔ Γ u+∂ ν a u+βu=g on ∂Ω generates a strongly continuous compact semigroup. Some analyticity results of the semigroup are also discussed.
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ABSTRACT Let A be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain Ω⊂ℝ N with Lipschitz continuous boundary ∂Ω, a realization of Au-β 1 (x,u) in C(Ω ¯) with the nonlinear general... more
ABSTRACT Let A be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain Ω⊂ℝ N with Lipschitz continuous boundary ∂Ω, a realization of Au-β 1 (x,u) in C(Ω ¯) with the nonlinear general Wentzell boundary conditions [Au-β 1 (x,u)]| ∂Ω -Δ Γ u+∂ ν a u+β 2 (x,u)=0 on ∂Ω generates a strongly continuous nonlinear semigroup on C(Ω ¯). Here, ∂ ν a u is the conormal derivative of u, and β 1 (x,·)(x∈Ω), β 2 (x,·)(x∈∂Ω) are continuous on ℝ satisfying a certain growth condition.
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Abstract We show that a realization of the Laplace operator Au:= u′′ with general nonlocal Robin boundary conditions α ju′(j)+ β ju (j)+ γ 1ju (1− j)= 0,(j= 0, 1) generates a cosine family on L p (0, 1) for every p ∈\, 1, ∞). Here α j, β... more
Abstract We show that a realization of the Laplace operator Au:= u′′ with general nonlocal Robin boundary conditions α ju′(j)+ β ju (j)+ γ 1ju (1− j)= 0,(j= 0, 1) generates a cosine family on L p (0, 1) for every p ∈\, 1, ∞). Here α j, β j and γ j are complex numbers ...
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Abstract In the first part of the paper, Gaussian estimates are used to study L^p-summability of the solution of the wave equation in L^p(Ω) associated with a general operator in divergence form with bounded coefficients. Secondly, we... more
Abstract In the first part of the paper, Gaussian estimates are used to study L^p-summability of the solution of the wave equation in L^p(Ω) associated with a general operator in divergence form with bounded coefficients. Secondly, we prove that if Ω is a cube in \RR^ ...
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Abstract Let Ω⊂ RN be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions∂ u∂ ν+ βu= 0 on the boundary∂ Ω generates a holomorphic C0-semigroup of angle... more
Abstract Let Ω⊂ RN be a bounded domain with Lipschitz boundary. We prove in the first part that a realization of the Laplacian with Robin boundary conditions∂ u∂ ν+ βu= 0 on the boundary∂ Ω generates a holomorphic C0-semigroup of angle π/2 on C (Ω) if 0< β0≤ β ...
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Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective,... more
Given an open set in , we prove that every function in is zero everywhere on the boundary if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mapping from into is injective, where denotes the Perron solution of the Dirichlet problem. Let be the set