1,559 research outputs found
A minimum problem with free boundary in Orlicz spaces
We consider the optimization problem of minimizing in the class of functions with
, for a given and bounded.
is the class of weakly differentiable functions with
. The conditions on the function G allow
for a different behavior at 0 and at . We prove that every solution u
is locally Lipschitz continuous, that they are solution to a free boundary
problem and that the free boundary, , is a regular
surface. Also, we introduce the notion of weak solution to the free boundary
problem solved by the minimizers and prove the Lipschitz regularity of the weak
solutions and the regularity of their free boundaries near
``flat'' free boundary points
Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes
The paper deals with the asymptotic behavior of solutions to a non-local
diffusion equation, , in an exterior domain, , which
excludes one or several holes, and with zero Dirichlet data on
. When the space dimension is three or more this
behavior is given by a multiple of the fundamental solution of the heat
equation away from the holes. On the other hand, if the solution is scaled
according to its decay factor, close to the holes it behaves like a function
that is -harmonic, , in the exterior domain and vanishes in its
complement. The height of such a function at infinity is determined through a
matching procedure with the multiple of the fundamental solution of the heat
equation representing the outer behavior. The inner and the outer behavior can
be presented in a unified way through a suitable global approximation
Asymptotic behavior for a nonlocal diffusion equation in exterior domains: the critical two-dimensional case
We study the long time behavior of bounded, integrable solutions to a
nonlocal diffusion equation, , where is a smooth,
radially symmetric kernel with support . The problem
is set in an exterior two-dimensional domain which excludes a hole
, and with zero Dirichlet data on . In the far field
scale, with , the scaled
function behaves as a multiple of the fundamental solution
for the local heat equation with a certain diffusivity determined by . The
proportionality constant, which characterizes the first non-trivial term in the
asymptotic behavior of the mass, is given by means of the asymptotic \lq
logarithmic momentum' of the solution,
. This asymptotic
quantity can be easily computed in terms of the initial data. In the near field
scale, with , the scaled
function converges to a multiple of , where is the unique stationary solution of the problem that
behaves as when . The proportionality constant is
obtained through a matching procedure with the far field limit. Finally, in the
very far field, with , the solution is
proved to be of order .Comment: 24 page
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