# Optimal design problems for the first $p-$fractional eigenvalue with mixed boundary conditions

## TL;DR

This paper investigates the optimal shape design for the first eigenvalue of the fractional p-Laplacian with mixed boundary conditions, establishing existence results and asymptotic bounds as the fractional parameter approaches 1.

## Contribution

It introduces a new optimal shape problem for fractional p-Laplacian eigenvalues with mixed boundary conditions, proving existence and analyzing asymptotic behavior.

## Key findings

- Existence of an optimal design for the shape problem.
- Asymptotic bounds for the eigenvalue as s approaches 1.
- Asymptotic bounds are independent of the measure constraint .

## Abstract

In this paper we study an optimal shape design problem for the first eigenvalue of the fractional $p-$laplacian with mixed boundary conditions. The optimization variable is the set where the Dirichlet condition is imposed (that is restricted to have measure equal than a prescribed quantity, $\alpha$). We show existence of an optimal design and analyze the asymptotic behavior when the fractional parameter $s\uparrow 1$ obtaining asymptotic bounds that are independent of $\alpha$.