We establish the existence and uniqueness theorem for the exterior Dirichlet problem for the special Lagrangian equation with prescribed asymptotic behavior at infinity, in both the viscosity setting for all the phases and classical setting for the critical and supercritical phases. These results generalize previous work by the second author by removing restrictive assumptions on the asymptotic matrix and improving the decay rate to the order $2-n$. We also solve the interior Dirichlet problem for the critical special Lagrangian equation and, as applications, all the above-mentioned corresponding problems for the three dimensional quadratic Hessian equation without any admissibility condition.
This paper studies the exterior Dirichlet problem for Hessian quotient equations with nonconstant right-hand sides. We prove the existence of classical admissible solutions with prescribed asymptotic Hessians and establish convergence of the Hessian at infinity. The main difficulties are obtaining second-order estimate...
We study Robin and Neumann problems for the scalar curvature equation of admissible graphs over bounded uniformly convex domains in three dimensions. Under a small-volume assumption, we prove existence and uniqueness for the Robin problem and obtain a classical Neumann solution as the Robin parameter tends to zero. The...
This paper presents a mathematical analysis of the inverse problem concerning the recovery of two unknown potential coefficients in a coupled Schrödinger equations system, within a bounded domain of the Euclidean space Rn with Dirichlet boundary conditions. Our analysis relies on Neumann boundary measurements, where we...
In this paper, we study an ill-posed initial-boundary value problem for a high-order inhomogeneous parabolic equation with one degeneracy line. We obtain a priori estimate of the solution of the problem, determine the correctness set, and prove the uniqueness and conditional stability of the solution in this set. As a...
I. Khajiev, E. Shobdarov· Uzbek Mathematical Journal· 0 citations
The aim of this work is to develop a self-contained existence and uniqueness theory for a doubly nonlinear parabolic problem with variable exponents and homogeneous Neumann boundary conditions. Our approach is based on Rothe's time-discretization method and builds on the results established by the author in a previous...
We investigate on the uniqueness property of the solutions to a parabolic problem with the data and the coefficient of zero order term only summable functions. Despite all this lack of regularity and without using any notion of entropy or renormalized solutions or the property to be a solution obtained by approximation...
Maria Michaela Porzio· Journal of Convex Analysis· 2 citations
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