| Abstract: |
| The classical semilinear problem
\[
-\Delta u = F(x,u) \qquad \text{in }\Omega\subset\mathbb{R}^N,
\]
has been widely studied when the nonlinearity $F$ is a given function.
However, several models in Physics and Mechanics naturally lead to
equations of the form \(-\Delta u = \mu(x,u),\) where $\mu$ is a Radon measure depending on the solution itself.
A relevant example is the interior Bernoulli free boundary problem,
in which the measure is supported on the unknown interface.
In the elliptic setting, D\`{\i}az--Padial--Rakotoson (2007) developed an
existence theory for such problems, showing that the Bernoulli condition
can be expressed through the identity
\[
\int_\Omega |\nabla u|^{p-2}\nabla u\cdot\nabla\varphi\,dx
= \int_{\partial(u^{-1}(1))} q\,\varphi\, d\mathcal{H}_{N-1},
\]
highlighting the role of the measure as a Lagrange multiplier on the
free boundary.
This work extends that framework to the \emph{evolutionary} setting.
A semi-implicit time discretization transforms the parabolic problem
into a sequence of elliptic problems of the previous type.
This scheme can be viewed as a backward Euler approximation of the
parabolic Bernoulli problem, where the regularization around the
Bernoulli level set yields diffuse approximations of the measure.
Passing to the limit provides a time-dependent Radon measure supported
on the evolving free boundary. Combining this approach with a
Ghoussoub--Preiss mountain pass argument yields existence of weak
solutions to the evolutionary Bernoulli free boundary problem. |
|