Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties of variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the k-th perimeter-normalized Steklov eigenvalue is $$8\pi k$$ 8 π k , which is the best upper bound for the $$k^{\text {th}}$$ k th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realizing a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $$k=1$$ k = 1 , the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

Segment Inequality and Almost Rigidity Structures for Integral Ricci Curvature

We will show the Cheeger–Colding segment inequality for manifolds with integral Ricci curvature bound. By using this segment inequality, the almost rigidity structure results for integral Ricci curvature will be derived by a similar method as in [1]. And the sharp Hölder continuity result of [7] holds in the limit space of manifolds with integral Ricci curvature bound.

Regularity results for a class of obstacle problems with p, q−growth conditions

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min{ ∫ΩF(x, Dz) : z ∈ 𝛫ψ(Ω)}. Here 𝛫ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝn, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.

Optimal harvesting of a hierarchical age-structured population system

We investigate an optimal harvesting problem for age-structured species, in which elder individuals are more competitive than younger ones, and the population is modeled by a highly nonlinear integro-partial differential equation with a global feedback boundary condition. The existence of optimal strategies is established by means of compactness and maximizing sequences, and the maximum principle obtained via an adjoint system, tangent-normal cones and a new continuity result. In addition, some numerical experiments are presented to show the effects of the price function and younger’s weight on the optimal profits.

THE DISTRIBUTIONAL -HESSIAN IN FRACTIONAL SOBOLEV SPACES

We introduce the notion of a distributional $k$-Hessian ($k=2,\ldots ,n$) associated with fractional Sobolev functions on $\unicode[STIX]{x1D6FA}$, a smooth bounded open subset in $\mathbb{R}^{n}$. We show that the distributional $k$-Hessian is weakly continuous on the fractional Sobolev space $W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$ and that the weak continuity result is optimal, that is, the distributional $k$-Hessian is well defined in $W^{s,p}(\unicode[STIX]{x1D6FA})$ if and only if $W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$.

Weighted Hölder continuity of Riemann-Liouville fractional integrals – Application to regularity of solutions to fractional cauchy problems with Carathéodory dynamics

This paper is dedicated to several original (weighted) Hölder continuity results for Riemann-Liouville fractional integrals of weighted integrable functions. As an application, we prove a new weighted continuity result for solutions to nonlinear Riemann-Liouville fractional Cauchy problems with Carathéodory dynamics.

An Ambrosetti–Prodi-type problem for the (p,q)-Laplacian operator

This work has objective to obtain results of existence and multiplicity of solutions for an Ambrosetti–Prodi-type problem for the [Formula: see text] operator. Moreover, it was proved a continuity result for the parameter which limits the existence of solutions in relation of the parameter [Formula: see text].

A continuity result on quadratic matings with respect to parameters of odd denominator rationals

In this paper we prove a continuity result on matings of quadratic lamination maps sp depending on odd denominator rationals p ∈(0,1). One of the two mating components is fixed in the result. Note that our result has its implication on continuity of matings of quadratic hyperbolic polynomials fc(z)=z2 + c, c ∈ M the Mandelbrot set with respect to the usual parameters c. This is because every quadratic hyperbolic polynomial in M is contained in a bounded hyperbolic component. Its center is Thurston equivalent to some quadratic lamination map sp, and there are bounds on sizes of limbs of M and on sizes of limbs of the mating components on the quadratic parameter slice Perm′(0).

Maximum Principle for Forward-Backward Stochastic Control System Driven by Lévy Process

We study a stochastic optimal control problem where the controlled system is described by a forward-backward stochastic differential equation driven by Lévy process. In order to get our main result of this paper, the maximum principle, we prove the continuity result depending on parameters about fully coupled forward-backward stochastic differential equations driven by Lévy process. Under some additional convexity conditions, the maximum principle is also proved to be sufficient. Finally, the result is applied to the linear quadratic problem.