Related papers: The Dirichlet problem without the maximum principl…

We prove a boundary Harnack inequality for nonlocal elliptic operators $L$ in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that if $Lu_1=Lu_2=0$ in $\Omega\cap B_1$, $u_1=u_2=0$ in…

We study the Dirichlet problem for systems of the form -\Delta u^k=f^k(x,u)+\mu^k, x\in\Omega, k=1,...,n, where \Omega\subset R^d$ is an open (possibly nonregular) bounded set, \mu^1,...,\mu^n are bounded diffuse measures on \Omega,…

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's…

Let $\Omega$ be an open, simply connected, and bounded region in $\mathbb{R}^{d}$, $d\geq2$, and assume its boundary $\partial\Omega$ is smooth. Consider solving the eigenvalue problem $Lu=\lambda u$ for an elliptic partial differential…

In this paper the existence of solutions, $(\lambda,u)$, of the problem $$-\Delta u=\lambda u -a(x)|u|^{p-1}u \quad \hbox{in }\Omega, \qquad u=0 \quad \hbox{on}\;\;\partial\Omega,$$ is explored for $0 < p < 1$. When $p>1$, it is known that…

We discuss the Dirichlet problem of the quasi-linear elliptic system \begin{eqnarray*} -e^{-f(U)}div(e^{f(U)}\bigtriangledown U)+&{1/2}f'(U)|\bigtriangledown U|^2&=0, {in $\Omega$}, & U|_{\partial\Omega}&=\phi. \end{eqnarray*} Here $\Omega$…

The boundary value problem is examined for the system of elliptic equations of from $-\Delta u + A(x)u = 0 \quad\text{in} \Omega,$ where $A(x)$ is positive semidefinite matrix on $\mathbb{R}^{{k}\times{k}},$ and $\frac{\partial u}{\partial…

The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions…

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge \delta > 0$ for all $x \in…

In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special…

This paper is devoted to analyse the Dirichlet problem for a nonlinear elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation arising in the level set formulation of the…

We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First,…

We discuss the existence and regularity of solutions to the following Dirichlet problem: $$\begin{equation} \begin{cases} -\textrm{div}\left(\frac{Du}{(1+|u|)^{\theta}}\right)= -\textrm{div}\left(u^{\gamma}E(x)\right)+f(x) \qquad & \mbox{in…

Let $L $ be a second order elliptic operator with smooth coefficients defined on a domain $\Omega \subset \mathbb{R}^d$ (possibly unbounded), $d\geq 3$. We study nonnegative continuous solutions $u$ to the equation $L u(x) - \varphi (x,…

We consider the Robin boundary value problem $\mathrm{div} (A \nabla u) = \mathrm{div} \mathbf{f}+F$ in $\Omega$, $\mathcal{C}^1$ domain, with $(A \nabla u - \mathbf{f})\cdot \mathbf{n} + \alpha u = g$ on $\Gamma$, where the matrix $A$…

We study uniqueness of Dirichlet problems of second order divergence-form elliptic systems with transversally independent coefficients on the upper half-space in absence of regularity of solutions. To this end, we develop a substitute for…

Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…

In this paper, we consider the following non-linear equations in unbounded domains $\Omega$ with exterior Dirichlet condition: \begin{equation*}\begin{cases} (-\Delta)_p^s u(x)=f(u(x)), & x\in\Omega,\\ u(x)>0, &x\in\Omega,\\ u(x)\leq0,…

For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary…

Assume that $p > 1$ and $p - 1 \le \alpha \le p$ are real numbers and $\Omega$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^\alpha \ge 0, $$ where $D =…