We variationally characterize the bottom of the spectrum of the buckling problem in an infinite cylinder A×ℝ^n-d, where A is an open bounded subset of ℝ^d, and compute it explicitly when A is a ball.

In this paper we study some relationships between the first Dirichlet Laplacian eigenvalue Λ (Ω ) and the torsional rigidity T(Ω ) of a domain Ω . We consider the problem of optimizing the product Λ (Ω )T(Ω ) among sets with prescribed perimeter, both in the class of open sets with finite perimeter and within the class of convex domains. We also characterize the values of q>0 for which the ball is a stable local minimizer or maximizer of the quantity Λ (Ω )T(Ω )^q , under either a volume or a perimeter constraint.

A celebrated inequality by Payne relates the first eigenvalue of the Dirichlet Laplacian to the first eigenvalue of the buckling problem. Motivated by the goal of establishing a quantitative version of this inequality, we sho that Payne's original estimate - which is not sharp - can in fact be improved. Our result provides a refined spectral bound and opens the way to further investigations into quantitative enhancements of classical inequalities in spectral theory.

In the present survey, we consider the classical reinforcement problem for elliptic boundary value problems originally studied by Sanchez-Palencia in 1969. We focus on the seminar papers by Brezis, Caffarelli, Friedman, and by Acerbi Buttazzo, and discuss the related optimisation problems proposed by Friedman and by Buttazzo.

In this paper, we establish a comparison principle in terms of Lorentz norms and pointwise inequalities between a positive solution u to the Poisson equation with non-homogeneous Neumann boundary conditions and a specific positive solution u to the Schwarz symmetrized problem, which is related to u through an additional boundary condition.

For every given β<0, we study the problem of maximizing the first Robin eigenvalue of the Laplacian λ_β(Ω) among convex (not necessarily smooth) sets Ω⊂𝕊^n with fixed perimeter. In particular, denoting by σ_n the perimeter of the n-dimensional hemisphere, we show that for fixed perimeters P<σ_n, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between Ω and the ball D of the same perimeter.

We study the behavior, as p -> + infinity {p\to+\infty} , of the second eigenvalues of the p-Laplacian with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that, up to some regularity of the set, the limit of the second eigenvalues is actually the second eigenvalue of the so-called infinity {\infty} -Laplacian.

We are interested in the thermal insulation of a bounded open set $\Omega$ surrounded by a set whose thickness is locally described by $\varepsilon h$, where $h$ is a non-negative function defined on the boundary $\partial\Omega$. We study the problem in the limit for $\varepsilon$ going to zero using a first-order asymptotic development by $\Gamma$-convergence.

In this paper, we prove a Serrin-type result for an elliptic system of equations, overdetermined with both Dirichlet and generalized Neumann conditions. With this tool, we characterize the critical shapes of some domain functionals under volume constraints.

In this paper, we introduce a symmetrization technique for the gradient of a $\BV$ function, which separates its absolutely continuous part from its singular part (sum of the jump and the Cantorian part). In particular, we prove an $\text{\emph{L}}^{\text{1}}$ comparison between the function and its symmetrized. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.

Abstract In this paper, we study the Γ-limit, as p → 1 {p\to 1} , of the functional J p ⁢ ( u ) = ∫ Ω | ∇ ⁡ u | p + β ⁢ ∫ ∂ ⁡ Ω | u | p ∫ Ω | u | p , J_{p}(u)=\frac{\int_{\Omega}\lvert\nabla u\rvert^{p}+\beta\int_{\partial\Omega% }\lvert u\rvert^{p}}{\int_{\Omega}\lvert u\rvert^{p}}, where Ω is a smooth bounded open set in ℝ N {\mathbb{R}^{N}} , p > 1 {p>1} and β is a real number. Among our results, for β > - 1 {\beta>-1} , we derive an isoperimetric inequality for Λ ⁢ ( Ω , β ) = inf u ∈ BV ⁡ ( Ω ) , u ≢ 0 ⁡ | D ⁢ u | ⁢ ( Ω ) + min ⁡ ( β , 1 ) ⁢ ∫ ∂ ⁡ Ω | u | ∫ Ω | u | \Lambda(\Omega,\beta)=\inf_{u\in\operatorname{BV}(\Omega),\,u\not\equiv 0}% \frac{\lvert Du\rvert(\Omega)+\min(\beta,1)\int_{\partial\Omega}\lvert u\rvert% }{\int_{\Omega}\lvert u\rvert} which is the limit as p → 1 + {p\to 1^{+}} of λ ⁢ ( Ω , p , β ) = min u ∈ W 1 , p ⁢ ( Ω ) ⁡ J p ⁢ ( u ) {\lambda(\Omega,p,\beta)=\min_{u\in W^{1,p}(\Omega)}J_{p}(u)} . We show that among all bounded and smooth open sets with given volume, the ball maximizes Λ ⁢ ( Ω , β ) {\Lambda(\Omega,\beta)} when β ∈ ( - 1 , 0 ) {\beta\in(-1,0)} and minimizes Λ ⁢ ( Ω , β ) {\Lambda(\Omega,\beta)} when β ∈ [ 0 , ∞ ) {\beta\in[0,\infty)} .

By means of a suitable weighted rearrangement, we obtain various apriori bounds for the solutions to a Robin problem. Among other things, we derive a family of Faber-Krahn type inequalities.

Comparison results of Talenti type for elliptic problems with Dirichlet boundary conditions have been widely investigated in recent decades. In this paper, we deal with Robin boundary conditions. Surprisingly, contrary to the Dirichlet case, Robin boundary conditions make the comparison sensitive to the dimension, and while the planar case seems to be completely settled, in higher dimensions some open problems are yet unsolved. © 2023 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Benedetto Croce (Pescasseroli 1866–Naples 1952) was a philosopher, historian, and literary critic and a prominent figure in Italian cultural and political life in the first half of the twentieth century whose thought had a significant international echo.

We study the thermal insulation of a bounded body omega G Double-struck capital R-n, under a prescribed heat source f > 0, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the environment determined by convection: this corresponds to Robin boundary conditions on the free boundary of the layer. We show that a minimal configuration exists and that it satisfies uniform density estimates.

Abstract We study the behaviour, when p → + ∞ p\to +\infty , of the first p-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞ \infty -Laplacian. Moreover, in the second part of the article, we focus our attention on the p-Poisson equation when the datum f f belongs to L ∞ ( Ω ) {L}^{\infty }\left(\Omega ) and we study the behaviour of solutions when p → ∞ p\to \infty .

Bernoulli free boundary problems (BFBP) govern many real applications, from chemistry to fluid dynamics. BFBP are overdetermined differential models in which the boundary of the solution's domain appears as an unknown. This work provides a variational formulation for BFBP, and a computational method focused on the novel methodology of Physics Informed Neural Networks (PINNs) is proposed. It consists in training a neural network to approximate the solution of the differential problem, minimizing a suitable cost function built, taking into account the physics constraint given by the model. In particular, since physics laws are injected through a loss function containing integral terms in our case, an approach based on quasi Monte-Carlo integration methods has been implemented. Moreover, an adaptive strategy to increase the model accuracy, exploiting topological information related to the shape of the solution, has been developed. Finally, when available, comparisons with the analytical solutions and studies in the case of non-convex domains assess the reliability of the PINN approach in the examined context.

In this paper we provide a comparison result between the solutions to the torsion problem for the Hermite operator with Robin boundary conditions and the one of a suitable symmetrized problem.

We study a shape optimization problem involving a solid $$K\subset {\mathbb {R}}^n$$ that is maintained at constant temperature and is enveloped by a layer of insulating material $$\Omega $$ which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all $$(K,\Omega )$$ with prescribed measure for K and $$\Omega $$ , and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on $$\partial \Omega $$ ) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.

In contemporary philosophical discourse the image of the hircocervus, rooted in the tradition of Western thought, is frequently invoked as a metaphor for the impossible and the incongruous. Not a few references to this legendary animal appear as well in the work of Benedetto Croce. One of these is at the centre of the investigation presented here: the uncertain interpretation of its meaning, widely misunderstood or even falsified, draws our attention to an important trait that runs through the history of Italian legal philosophy in the twentieth century. It is around this trait that the discussion in this essay develops.