Quasilinear parabolic equations with nonlinear Wentzell-Robin type boundary conditions

Let Ω⊂RN be a bounded domain with Lipschitz boundary, with a>0 on . Let σ be the restriction to ∂Ω of the (N−1)-dimensional Hausdorff measure and let be σ-measurable in the first variable and assume that for σ-a.e. x∈∂Ω, B(x,⋅) is a proper, convex, lower semicontinuous functional. We prove in the first part that for every p∈(1,∞), the operator Ap:=div(a|∇u|^p−2∇u) with nonlinear Wentzell-Robin type boundary conditions

     

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H¹-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))_t?0 on L²(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p?2 and every t>0, one has

     

Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if

     

Singularly perturbed nonlinear Neumann problems with a general nonlinearity

Let Ω be a bounded domain in Rn, n?3, with the boundary ∂Ω∈C³. We consider the following singularly perturbed nonlinear elliptic problem on Ω

     

Heat equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the heat equation posed in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

Boundary singularities for weak solutions of semilinear elliptic problems

Let Ω be a bounded domain in RN, N?2, with smooth boundary ∂Ω. We construct positive weak solutions of the problem Δu+up=0 in Ω, which vanish in a suitable trace sense on ∂Ω, but which are singular at prescribed isolated points if p is equal or slightly above . Similar constructions are carried out for solutions which are singular at any given embedded submanifold of ∂Ω of dimension k∈[0,N−2], if p equals or it is slightly above , and even on countable families of these objects, dense on a given closed set. The role of the exponent (first discovered by Brezis and Turner [H. Brezis, R. Turner, On a class of superlinear elliptic problems, Comm. Partial Differential Equations 2 (1977) 601-614]) for boundary regularity, parallels that of for interior singularities.     

On the Laplace equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the Laplace equation in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

Exact asymptotic behavior near the boundary to the solution for singular nonlinear Dirichlet problems

We take up the existence and the exact asymptotic behavior near the boundary ∂Ω of the unique classical solution to a singular Dirichlet problem

     

Solutions to critical elliptic equations with multi-singular inverse square potentials

Let Ω be an open-bounded domain in R^N(N?3) with smooth boundary ∂Ω. We are concerned with the multi-singular critical elliptic problem

     

Multiple fixed-sign solutions for a system of generalized right focal problems with deviating arguments

We consider the following system of generalized right focal boundary value problems

     

Existence of positive solutions for the Hénon equation involving critical Sobolev terms

Let N?3, ^2*=2N/(N−2) and Ω⊂RN be a bounded domain with a smooth boundary ∂Ω and 0∈Ω. Our purpose in this paper is to consider the existence of solutions of Hénon equation:

     

Up-to boundary regularity for a singular perturbation problem of p-Laplacian type

In this paper we are interested in establishing up-to boundary uniform estimates for the one phase singular perturbation problem involving a nonlinear singular/degenerate elliptic operator. Our main result states: if Ω⊂Rn is a C1,α domain, for some 0<α<1 and uε verifies

     

Exponential energy decay of solutions for an integro-differential equation with strong damping

The initial boundary value problem for an integro-differential equation with strong damping in Ω×(0,∞):

     

A free boundary problem for p-Laplacian in the plane

We consider the following free boundary problem in an unbounded domain Ω in two dimensions: Δpu=0 in Ω, on J₀, on J₁, where ∂Ω=J₀∪J₁. We prove that if 0<u<1 in Ω, Ji is the graph of a function in and gi is a constant for each i=0,1, then the free boundary ∂Ω must be two parallel straight lines and the solution u must be a linear function. The proof is based on maximum principle.     

Mixed boundary value problem of Laplace equation in a bounded Lipschitz domain

We study the existence and uniqueness of the mixed boundary value problem for Laplace equation in a bounded Lipschitz domain Ω⊂Rn, n?3. Let the boundary ∂Ω of Ω be decomposed by , Γ₁∩Γ₂=∅. We will show that if the Neumann data ψ is in and the Dirichlet data f is in , then the mixed boundary value problem has a unique solution and the solution is represented by potentials.     

Cauchy transform on nonrectifiable surfaces in Clifford analysis

The problem of reconstructing a monogenic Clifford algebra valued function on the boundary Γ of a general open set Ω in Rn+1 from a prescribed jump data u over the boundary is deeply connected with the study of the Clifford-Cauchy transform

     

Removable singularities for a Sobolev space

Let Ω⊂RN be an open set and F a relatively closed subset of Ω. We show that if the (N−1)-dimensional Hausdorff measure of F is finite, then the spaces and coincide, that is, F is a removable singularity for . Here is the closure of in H¹(Ω) and H¹(Ω) denotes the first order Sobolev space. We also give a relative capacity criterium for this removability. The space is important for defining realizations of the Laplacian with Neumann and with Robin boundary conditions. For example, if the boundary of Ω has finite (N−1)-dimensional Hausdorff measure, then our results show that we may replace Ω by the better set (which is regular in topology), i.e., Neumann boundary conditions (respectively Robin boundary conditions) on Ω and on coincide.     

Existence and iteration of positive solutions for a generalized right-focal boundary value problem with p-Laplacian operator

In the paper, we obtain the existence of positive solutions and establish a corresponding iterative scheme for the following third-order generalized right-focal boundary value problem with p-Laplacian operator:

     

The attractors for the nonhomogeneous nonautonomous Navier-Stokes equations

In this paper, we consider the attractors for the two-dimensional nonautonomous Navier-Stokes equations in nonsmooth bounded domain Ω with nonhomogeneous boundary condition u=φ on ∂Ω. Assuming , which is translation compact and φ∈L^∞(∂Ω), we establish the existence of the uniform attractor in L²(Ω) and .