Related papers: Distinct solutions to generated Jacobian equations…

We consider the heat equation in a smooth bounded convex domain $\Omega \subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_\nu u = \lambda (u - u^3)$. Stable non-constant stationary solutions do not exist when…

For multispecies ions, we study boundary layer solutions of charge conserving Poisson-Boltzmann (CCPB) equations [50] (with a small parameter \k{o}) over a finite one-dimensional (1D) spatial domain, subjected to Robin type boundary…

Let $\Omega \subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(\Omega)^2$ for every $f\in L^p_0(\Omega)$, if…

In the present paper we prove uniqueness results for solutions to a class of Neumann boundary value problems whose prototype is --div((1 + |$\nabla$u| 2) (p--2)/2 $\nabla$u) -- div(c(x)|u| p--2 u) = f in $\Omega$, (1 + |$\nabla$u| 2)…

Let $(u_\varepsilon)$ be a family of solutions of the Ginzburg--Landau equation with boundary condition $u_\varepsilon = g$ on $\partial \Omega$ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0 = g$ on $\partial \Omega$.…

We establish gradient higher integrability results for weak solutions to degenerate parabolic equations of double phase type $$ u_t-\operatorname{div} \left(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du\right)=0 $$ in $\Omega_T := \Omega\times (0,T)$,…

We consider the eigenvalue problem for the {\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\Omega \subset \mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues…

We present a constructive method to devise boundary conditions for solutions of second-order elliptic equations so that these solutions satisfy specific qualitative properties such as: (i) the norm of the gradient of one solution is bounded…

A classical result about unit equations says that if $\Gamma_1$ and $\Gamma_2$ are finitely generated subgroups of $\mathbb C^\times$, then the equation $x+y=1$ has only finitely many solutions with $x\in\Gamma_1$ and $y\in \Gamma_2$. We…

We consider a bounded open subset $\Omega$ of ${\mathbb{R}}^n$ of class $C^{1,\alpha}$ for some $\alpha\in]0,1[$ and we solve the Neumann problem for the Helmholtz equation both in $\Omega$ and in the exterior of $\Omega$. We look for…

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N+1}$ with a connected $C^{2,\epsilon}$ ($\epsilon\in(0,1)$) boundary. We show that, if the following overdetermined elliptic problem \begin{equation} -\Delta u=\alpha u\,\,…

In this paper, we perform a comparison study of two methods (the embedded boundary method and several versions of the mixed finite element method) to solve an elliptic boundary value problem.

We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$, known as general boundary-value problems. Their solutions $y:[a,b]\to \mathbb{C}^{m}$ belong to the Sobolev space…

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to $W^{1,p}(\Omega)$.…

In \cite{CJ1} M. Jaoua et al. studied the linear approximation of Robin problem on $\Omega$ an open bounded domain of $\R^d$, and they given some important results. In this paper, we study a nonlinear approximation of an elliptic problem…

In this paper, we prove the boundary pointwise $C^{0}$-regularity of weak solutions for Dirichlet problem of elliptic equations in divergence form with distributional coefficients, where the boundary value equals to zero. This is a…

We demonstrate a kind of linear superposition for a large number of nonlinear equations, both continuum and discrete. In particular, we show that whenever a nonlinear equation admits solutions in terms of Jacobi elliptic functions…

We prove that the uniqueness results obtained in \cite{urrea} for the Benjamin equation, cannot be extended for any pair of non-vanishing solutions. On the other hand, we study uniqueness results of solutions of the Benjamin equation. With…

We prove the joints conjecture, showing that for any $N$ lines in ${\Bbb R}^3$, there are at most $O(N^{{3 \over 2}})$ points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given $N^2$ lines…

We prove the existence and $C^{1,\alpha}$ regularity of solutions to nonlocal fully nonlinear elliptic double obstacle problems. We also obtain boundary regularity for these problems. The obstacles are assumed to be Lipschitz…