Related papers: Eigenvalue and Dirichlet problem for fully-nonline…
We use an iteration procedure propped up by a a classical form of the maximum principle to show the existence of solutions to a nonlinear Poisson equation with Dirichlet boundary conditions. These methods can be applied to the case of…
We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the $p$-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal…
We consider the Dirichlet problem for a class of elliptic and parabolic equations in the upper-half space $\mathbb{R}^d_+$, where the coefficients are the product of $x_d^\alpha, \alpha \in (-\infty, 1),$ and a bounded uniformly elliptic…
We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and…
We compute estimates for eigenvalues of a class of linear second-order elliptic differential operators in divergence form (with Dirichlet boundary condition) on a bounded domain in a complete Riemannian manifold. Our estimates are based…
The paper makes use of recent results in the theory of Banach lattices and positive operators to deal with abstract semilinear equations. The aim is to work with minimal or no regularity conditions on the boundary of the domains, where the…
In this paper, our main goal is to study the evolution problem associated with the Laplacian operator with Dirichlet boundary conditions on a regular tree. To this end, we place special emphasis on the associated first eigenvalue problem,…
In this paper, we use a probabilistic approach to show that there exists a unique, bounded continuous solution to the Dirichlet boundary value problem for a general class of second order non-symmetric elliptic operators $L$ with singular…
In this paper we survey some results on the Dirichlet problem \[\left\{ \begin{array}{rcll} L u &=&f&\textrm{in }\Omega \\ u&=&g&\textrm{in }\mathbb R^n\backslash\Omega \end{array}\right.\] for nonlocal operators of the form…
In this manuscript, we investigate a priori estimates for the solution to the Dirichlet eigenvalue problem for a broad class of concave elliptic Hessian operators of the form \[ F(D^2u)=-\Lambda u \quad \textrm{in} \, \Omega, \qquad u=0…
We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are…
In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong…
In this paper we study the Dirichlet eigenvalue problem $$ -\Delta_p u-\Delta_{J,p}u =\lambda|u|^{p-2}u \quad \text{ in } \Omega,\quad u=0 \quad\text{ in } \Omega^c=\mathbb{R}^N\setminus\Omega. $$ Here $\Delta_p u$ is the standard local…
We investigate strong maximum (and minimum) principles for fully nonlinear second order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of…
In this paper we prove existence and uniqueness of viscosity solutions of elliptic systems associated to fully nonlinear operators for minimization problems that involve interconnected obstacles. This system appears, among other, in the…
This paper studies nonlinear eigenvalues problems with a double non homogeneity governed by the $p(x)$-Laplacian operator, under the Dirichlet boundary condition on a bounded domain of $\mathbb{R}^N(N\geq2)$. According to the type of the…
Second order nonlinear eigenvalue problems are considered for which the spectrum is an interval. The boundary conditions are of Robin and Dirichlet type. The shape and the number of solutions are discussed by means of a phase plane…
We consider the nonlinear eigenvalue problem $ L u = \lambda f(u) $, posed in a smooth bounded domain $ \Omega \subseteq \Bbb{R}^{N} $ with Dirichlet boundary condition, where $ L $ is a uniformly elliptic second-order linear differential…
In this paper we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue $\lambda_{F}(p,\Omega)$ of the anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means of the $\mathcal…
We study the validity of the comparison and maximum principles, and their relation with principal eigenvalues, for a class of degenerate nonlinear operators that are extremal among operators with one dimensional fractional diffusion.