| Abstract: |
| Our aim is to study the existence of solutions for a nonlocal evolution
interior Bernoulli-type free boundary problem with a unknown measure data.
We observed that semilinear problems that can be written as $-\Delta
u(x)=F\left( x,u(x)\right) ,\ x\in\Omega$ (are given open bounded set in $%
\mathbb{R}^{N}$), with boundary conditions, are intensively studied in the
literature when $F$ is a function from $\Omega\times\mathbb{R}$ into $%
\mathbb{R}$. But, we realize that some models in Physics, or Mechanics can
be expressed as $-\Delta u(x)=\mu(x,u)$ in $\mathcal{D}^{\prime}(\Omega),$
where $\mu(x,u)$ is a Radon measure depending also on the own solution $u$.
The problem arises in several nonlinear flow laws and physical situation.
The elliptic problem was studied by J.I. D\`{\i}az, J.F. Padial and J.M.
Rakotoson in `On some Bernoulli free boundary type problems for general
elliptic operators`, Proceedings of the Royal Society of Edinburgh, 137A
(2007), 895-911. To look for a weak solution to the associated evolution
problem, we introduce a semi-implicit time differencing in order to obtain a
family of elliptic problems. For each one of this problems, we find weak
solution by applying a general mountain pass principle due to Ghoussoub-
Preiss for a sequence of approximate nonsingular problems. Finally, apriori
estimates allow us to obtain the solution by passing to the limit. (Joint work with J.M. Rakotoson). |
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