A Distributed Parabolic Control with Mixed Boundary Conditions

We study the asymptotic behavior of an optimal distributed control problem where the state is given by the heat equation with mixed boundary conditions. The parameter α intervenes in the Robin boundary condition and it represents the heat transfer coefficient on a portion Γ1 of the boundary of a given regular n-dimensional domain. For each α, the distributed parabolic control problem optimizes the internal energy g. It is proven that the optimal control ĝα with optimal state uĝαα and optimal adjoint state pĝαα are convergent as α → 1 (in norm of a suitable Sobolev parabolic space) to ĝ, uĝ and pĝ, respectively, where the limit problem has Dirichlet (instead of Robin) boundary conditions on Γ1. The main techniques used are derived from the parabolic variational inequality theory

Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).Comment: 26 pages, 20 figure

On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity

Eigenvalue asymptotic of Robin Laplace operators on two-dimensional domains with cusps

We consider Robin Laplace operators on a class of two-dimensional domains with cusps. Our main results include the formula for the asymptotic distribution of the eigenvalues of such operators. In particular, we show how the eigenvalue asymptotic depends on the geometry of the cusp and on the boundary conditions

On the pp-Laplacian with Robin boundary conditions and boundary trace theorems

Let $\Omega\subset\mathbb{R}^\nu$, $\nu\ge 2$, be a $C^{1,1}$ domain whose boundary $\partial\Omega$ is either compact or behaves suitably at infinity. For $p\in(1,\infty)$ and $\alpha>0$, define $$\Lambda(\Omega,p,\alpha):=\inf_{\substack{u\in W^{1,p}(\Omega)\\ u\not\equiv 0}}\dfrac{\displaystyle \int_\Omega |\nabla u|^p \mathrm{d} x - \alpha\displaystyle\int_{\partial\Omega} |u|^p\mathrm{d}\sigma}{\displaystyle\int_\Omega |u|^p\mathrm{d} x},$$ where $\mathrm{d}\sigma$ is the surface measure on $\partial\Omega$. We show the asymptotics $$\Lambda(\Omega,p,\alpha)=-(p-1)\alpha^{\frac{p}{p-1}} - (\nu-1)H_\mathrm{max}\, \alpha + o(\alpha), \quad \alpha\to+\infty,$$ where $H_\mathrm{max}$ is the maximum mean curvature of $\partial\Omega$. The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities