Related papers: Existence of free boundaries for some overdetermin…
We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the…
In this paper, we analyze an eigenvalue problem for nonlinear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We prove bifurcation results from trivial solutions and from infinity for…
We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential…
The goal of this paper is to derive in the two-dimensional case necessary and sufficient minimality conditions in terms of the second variation for the functional \[ v\mapsto\int_{\Omega}\big(|\nabla v|^{2}+\chi_{\{v>0\}}Q^{2} \big)\,dx, \]…
The aim of the paper is firstly to study domains of definitions in terms of boundary conditions of minimal and maximal operators, as well as selfadjoint extensions of a minimal operator associated with the fourth-order differential operator…
In this work, we study the existence and multiplicity of solutions for a class of problems involving the $\phi$-Laplacian operator in a bounded domain, where the nonlinearity has a critical growth. The main tool used is the variational…
An acute look at \underbar{basic} facts concerning \underbar{unbounded} subnormal operators is taken here. These operators have the richest structure and are the most exciting among the whole family of beneficiaries of the normal ones.…
We prove that flat or Lipschitz free boundaries of two-phase free boundary problems governed by fully nonlinear uniformly elliptic operators and with non-zero right hand side are $C^{1,\gamma}$.
In this article, we consider the boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence…
The present paper is mainly concerned with equations involving exponentials of bounded normal operators. Conditions implying commutativity of those normal operators are given. This is carried out without the known $2\pi i$-congruence-free…
The paper develops a theory of spectral boundary value problems from the perspective of general theory of linear operators in Hilbert spaces. An abstract form of spectral boundary value problem with generalized boundary conditions is…
The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential…
In this note we consider weighted conditional type operators between different Orlicz spaces and generalized conditional type Holder inequality that we defined in [2]. Then we give some necessary and sufficient conditions for boundedness of…
We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds $M$ of finite volume. Sharp conditions ensuring $L^q(M)$ and $L^\infty (M)$ bounds for eigenfunctions are exhibited in terms of either the isoperimetric…
On a bounded Lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to Robin boundary conditions, where the coefficients in the boundary conditions are functions.…
Bernoulli's free boundary problem is an overdetermined problem in which one seeks an annular domain such that the capacitary potential satisfies an extra boundary condition. There exist two different types of solutions called elliptic and…
We turn back to some pioneering results concerning, in particular, nonlinear potential theory and non-homogeneous boundary value problems for the so called p-Laplacian operator. Unfortunately these results, obtained at the very beginning of…
In this work, we study the asymptotic behavior of the free boundary of the solution to the exterior Bernoulli problem for the half Laplacian when the Bernoulli's gradient parameter tends to $0^+$ and to $+\infty$. Moreover, we show that,…
One method to determine whether or not a system of partial differential equations is consistent is to attempt to construct a solution using merely the "algebraic data" associated to the system. In technical terms, this translates to the…
We deal with a linear hyperbolic differential operator of the second order on a bounded planar domain with a smooth boundary. We establish a well-posedness result in case where a mixed, Dirichlet-Neumann, condition is prescribed on the…