607 research outputs found
Boundary regularity for viscosity solutions of fully nonlinear elliptic equations
We provide regularity results at the boundary for continuous viscosity
solutions to nonconvex fully nonlinear uniformly elliptic equations and
inequalities in Euclidian domains. We show that (i) any solution of two sided
inequalities with Pucci extremal operators is on the boundary;
(ii) the solution of the Dirichlet problem for fully nonlinear uniformly
elliptic equations is on the boundary; (iii) corresponding
asymptotic expansions hold. This is an extension to viscosity solutions of the
classical Krylov estimates for smooth solutions.Comment: 24 page
Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities
We study fully nonlinear elliptic equations such as in or in exterior domains, where is any uniformly elliptic,
positively homogeneous operator. We show that there exists a critical exponent,
depending on the homogeneity of the fundamental solution of , that sharply
characterizes the range of for which there exist positive supersolutions
or solutions in any exterior domain. Our result generalizes theorems of
Bidaut-V\'eron \cite{B} as well as Cutri and Leoni \cite{CL}, who found
critical exponents for supersolutions in the whole space , in case
is Laplace's operator and Pucci's operator, respectively. The arguments we
present are new and rely only on the scaling properties of the equation and the
maximum principle.Comment: 16 pages, new existence results adde
Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems
We study qualitative properties of positive solutions of noncooperative,
possibly nonvariational, elliptic systems. We obtain new classification and
Liouville type theorems in the whole Euclidean space, as well as in
half-spaces, and deduce a priori estimates and existence of positive solutions
for related Dirichlet problems. We significantly improve the known results for
a large class of systems involving a balance between repulsive and attractive
terms. This class contains systems arising in biological models of
Lotka-Volterra type, in physical models of Bose-Einstein condensates and in
models of chemical reactions.Comment: 35 pages, to appear in Archive Rational Mech. Ana
Nonexistence of positive supersolutions of elliptic equations via the maximum principle
We introduce a new method for proving the nonexistence of positive
supersolutions of elliptic inequalities in unbounded domains of .
The simplicity and robustness of our maximum principle-based argument provides
for its applicability to many elliptic inequalities and systems, including
quasilinear operators such as the -Laplacian, and nondivergence form fully
nonlinear operators such as Bellman-Isaacs operators. Our method gives new and
optimal results in terms of the nonlinear functions appearing in the
inequalities, and applies to inequalities holding in the whole space as well as
exterior domains and cone-like domains.Comment: revised version, 32 page
Landesman-Lazer type results for second order Hamilton-Jacobi-Bellman equations
We extend the classical Landesman-Lazer results to the setting of second
order Hamilton-Jacobi-Bellman equations. A number of new phenomena appear
Solvability of nonlinear elliptic equations with gradient terms
We study the solvability in the whole Euclidean space of coercive
quasi-linear and fully nonlinear elliptic equations modeled on , , where and are increasing continuous
functions. We give conditions on and which guarantee the availability
or the absence of positive solutions of such equations in . Our results
considerably improve the existing ones and are sharp or close to sharp in the
model cases. In particular, we completely characterize the solvability of such
equations when and have power growth at infinity. We also derive a
solvability statement for coercive equations in general form
Fundamental solutions of homogeneous fully nonlinear elliptic equations
We prove the existence of two fundamental solutions and
of the PDE for
any positively homogeneous, uniformly elliptic operator . Corresponding to
are two unique scaling exponents which
describe the homogeneity of and . We give a sharp
characterization of the isolated singularities and the behavior at infinity of
a solution of the equation , which is bounded on one side. A
Liouville-type result demonstrates that the two fundamental solutions are the
unique nontrivial solutions of in
which are bounded on one side in a neighborhood of the origin as well as at
infinity. Finally, we show that the sign of each scaling exponent is related to
the recurrence or transience of a stochastic process for a two-player
differential game.Comment: 35 pages, typos and minor mistakes correcte
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