Corpus ID: 237940232

How to construct parametrized families of free boundaries near nondegenerate solutions

@inproceedings{Cavallina2021HowTC,
  title={How to construct parametrized families of free boundaries near nondegenerate solutions},
  author={Lorenzo Cavallina},
  year={2021}
}
In this paper, we introduce the notion of variational free boundary problem. Namely, we say that a free boundary problem is variational if its solutions can be characterized as the critical points of some shape functional. Moreover, we extend the notion of nondegeneracy of a critical point to this setting. As a result, we provide a unified functional-analytical framework that allows us to construct families of solutions to variational free boundary problems whenever the shape functional is… Expand

Figures from this paper

References

SHOWING 1-10 OF 26 REFERENCES
On the qualitative theory of parametrized families of free boundaries.
The literature on free boundary problems usually focusses on the individual Solutions (their existence, regularity, geometric properties, etc.). Our purpose here is to study parametrized families ofExpand
Existence and stability of solutions for a fourth order overdetermined problem
Abstract We examine a Serrin-type overdetermined boundary value problem for the biharmonic operator. If the underlying set is the unit ball, a solution exists for a constant overdeterminingExpand
Structure of shape derivatives
Abstract. In this paper, we describe the precise structure of second "shape derivatives", that is derivatives of functions whose argument is a variable subset of $ \mathbb{R}^N $. This is done forExpand
Structure of shape derivatives around irregular domains and applications
In this paper, we describe the structure of shape derivatives around sets which are only assumed to be of finite perimeter in $\R^N$. This structure allows us to define a useful notion of positivityExpand
Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the EuclideanExpand
Bernoulli's free-boundary problem, qualitative theory and numerical approximation.
Bernoulli's free-boundary problem arises in ideal fluid dynamics, optimal insulation and electro chemistry. In electrostatic terms we design an annular condenser with a prescribed and an unknownExpand
A symmetry problem in potential theory
The proof of this result is given in Section 1 ; in Section 3 we give various generalizations to elliptic differential equations other than (1). Before turning to the detailed arguments it will be ofExpand
An overdetermined problem with non-constant boundary condition
We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shapeExpand
On a two-phase Serrin-type problem and its numerical computation
We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability nearExpand
Stability analysis of the two-phase torsional rigidity near a radial configuration
ABSTRACT Let denote the unit ball of () centered at the origin. We suppose that contains a core, given by a smaller concentric ball , made of a (possibly) different material. We discover that,Expand
...
1
2
3
...