2,372 research outputs found
To the theory of semi-linear equations in the plane
In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z),whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak C(Ď )∩ W¹,² loc (D) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semi-linear equations of mathematical physics, arising from modelling processes in anisotropic and inhomogeneous media. With a view to further development of the theory of boundary value problems for the semi-linear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity
Oblique boundary value problems for augmented Hessian equations I
In this paper, we study global regularity for oblique boundary value problems
of augmented Hessian equations for a class of general operators. By assuming a
natural convexity condition of the domain together with appropriate convexity
conditions on the matrix function in the augmented Hessian, we develop a global
theory for classical elliptic solutions by establishing global a priori
derivative estimates up to second order. Besides the known applications for
Monge-Amp`ere type operators in optimal transportation and geometric optics,
the general theory here embraces prescribed mean curvature problems in
conformal geometry as well as oblique boundary value problems for augmented
k-Hessian, Hessian quotient equations and certain degenerate equations.Comment: Revised version containing minor clarification
Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions
Assuming is a ball in , we analyze the positive
solutions of the problem that branch out from the constant solution as grows from to
. The non-zero constant positive solution is the unique positive
solution for close to . We show that there exist arbitrarily many
positive solutions as (in particular, for supercritical exponents)
or as for any fixed value of , answering partially a
conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for
and so that a given number of solutions exist. The geometrical properties
of those solutions are studied and illustrated numerically. Our simulations
motivate additional conjectures. The structure of the least energy solutions
(among all or only among radial solutions) and other related problems are also
discussed.Comment: 37 pages, 24 figure
Entire subsolutions of fully nonlinear degenerate elliptic equations
We prove existence and non existence results for fully nonlinear degenerate
elliptic inequalities, by showing that the classical Keller--Osserman condition
on the zero order term is a necessary and sufficient condition for the
existence of entire sub solutions
Diffusive Logistic Equations with Harvesting and Heterogeneity Under Strong Growth Rate
We consider the equation −Δu=au−b(x)u2−ch(x) in Ω,u=0 on ∂Ω, where Ω is a smooth bounded domain in RN, b(x) and h(x) are nonnegative functions, and there exists Ω0⊂⊂Ω such that {x:b(x)=0}=Ω¯¯¯0. We investigate the existence of positive solutions of this equation for c large under the strong growth rate assumption a≥λ1(Ω0), where λ1(Ω0) is the first eigenvalue of the −Δ in Ω0 with Dirichlet boundary condition. We show that if h≡0 in Ω∖Ω¯¯¯0, then our equation has a unique positive solution for all c large, provided that a is in a right neighborhood of λ1(Ω0). For this purpose, we prove and utilize some new results on the positive solution set of this equation in the weak growth rate case
Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators
We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators. We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators
Concentration of Solutions for a Singularly Perturbed Neumann Problem in non smooth domains
We consider the equation in a bounded
domain with edges. We impose Neumann boundary conditions,
assuming , and prove concentration of solutions at suitable points of
on the edges.Comment: 24 pages. Second Version, minor changes. To appear in Annales de
l'Institut Henri Poincar\'e - Analyse non lin\'eair
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