Shape optimization of a weighted two-phase Dirichlet eigenvalue

Computational approaches for extremal geometric eigenvalue problems

On some p(x)-quasilinear problem with right-hand side measure

We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ⁢ ( Ω ) {E_{f}(\Omega)} of the Laplacian in the domain Ω or the first eigenvalue λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other natural constraints), we instead consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate their convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional was usually the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f ⁢ ( Ω ) {E_{f}(\Omega)} or λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} when shapes are smooth and convex. The main novelty in the present paper is the proof of a weak convexity property of E f ⁢ ( Ω ) {E_{f}(\Omega)} and λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} among planar convex shapes, namely rather nonsmooth shapes. This involves new computations and estimates of the second order shape derivatives of E f ⁢ ( Ω ) {E_{f}(\Omega)} and λ 1 ⁢ ( Ω ) {\lambda_{1}(\Omega)} interesting for themselves.

By analyzing variational problems formulated in the Unified Form Language, a structure-aware differentiation tool is presented which can automatically generate both the classical boundary representation and the weak or “volume” formulation of first and second order shape derivatives. Where applicable, the tool can either automatically apply the divergence theorem in tangent spaces for the strong form or calculate discrete material derivatives for the weak form. Furthermore, additional assumptions and simplifications can also be automatically applied, such that a repeated application leads to symmetric shape Hessians. The resulting expression can then be processed by the FEniCS environment, resulting in the semiautomatic creation of shape optimization chains from a user-supplied Lagrangian only. The methodology is tested by conducting shape Newton optimization using examples from geometry and CFD. The respective software is released as open source, available from https://bitbucket.org/Epoxid/femorph.

Our goal is to introduce a Newton method in computing the stationary points of a total energy with respect to the shape. We formulated a precise description of the second order shape derivative. It is given by a symmetrical boundary integral operator, useful for numerical calculations. This method is applied to a particular shape optimisation problem, the electromagnetic casting problem.

This work deals with theH 1 condition numbers and the distribution of theB h singular values of the preconditioned operators {B h ?1 A h }0<h<1, whereA h andB h are finite element discretizations of second order elliptic operators,A andB respectively.B is also assumed to be self-adjoint and positive definite. For conforming finite elements, Parter and Wong have shown that the singular values "cluster" in a positive finite interval. Goldstein also has derived results on the spectral distribution ofB h ?1 A h using a different approach. As a generalization of the results of Parter and Wong, the current work includes nonconforming finite element methods which deal with Dirichlet boundary conditions. It will be shown that, in this more general setting, the singular values also "cluster" in a positive finite interval. In particular, if the leading part ofB is the same as the leading part ofA, then the singular values cluster about the point {1}. Two specific methods are given as applications of this theory. They are the penalty method of Babuska and the method of "nearly zero" boundary conditions of Nitsche. Finally, it will be shown that the same results can be proven by an approach generalized from the work of Goldstein.

In the present paper, in view of the topological degree methods and the theory of the variable exponent Sobolev spaces, we discuss a Dirichlet boundary value problem for elliptic equations involving the $(p(x),q(x))$-Laplacian operator with a reaction term depending on the gradient and on two real parameters. Under certain assumptions, we establish the existence of at least one weak solution to this problem. Our results extends some recent work in the literature.

We study the existence and multiplicity of positive solutions to $n\times n$ systems of the form \begin{align*} -\Delta u_1&=\lambda f_1(u_2)& \mbox{ in }\Omega\\ -\Delta u_2&=\lambda f_2(u_3)&\mbox{ in }\Omega\\ \vdots \quad &= \quad\vdots&\\ -\Delta u_{n-1}&=\lambda f_{n-1}(u_n)&\mbox{ in }\Omega\ \ -\Delta u_n&=\lambda f_n(u_1)&\mbox{ in }\Omega\\ u_1&=u_2=...=u_n=0 &\mbox{ on }\partial\Omega. \end{align*} Here $\Delta$ is the Laplacian operator, $ \lambda$ is a non-negative parameter, $\Omega$ is a bounded domain in $\mathbb R^N$ with smooth boundary $\partial\Omega$ and $f_i\in C^1([0,\infty)),$ $i\in\{1,2,\dots,n\},$ belongs to a class of strictly increasing functions that have a combined sublinear effect at $\infty$. We establish results for positone systems ($f_i(0)\geq0,$ $ i\in\{1,\dots,l-1,l+1,\dots,n\}$ and $f_l(0)>0$ for some $l\in\{1,\dots,n\}$), semipositone systems (no sign conditions on $f_i(0)$) and for systems with $f_i(0)=0,$ $ i\in\{1,2,\dots,n\}.$ We establish our results by the method of sub and supersolutions.

In this paper we study the second-order shape derivative for the solution to the wave equation with Dirichlet boundary conditions. We first prove the shape derivative exists for this problem. Specific complications arise when differentiating with respect to the domain hyperbolic PDEs; this requires an exclusive approach using the hidden regularity described in [7]. This study has been started in [1] and announced in [2]. We present here a new result where the data are no longer defined on the whole D and then restricted to the subsets of D but rather defined differently on each domain of D. Moreover, the differentiability is improved under stronger regularity of those functions. The Dirichlet boundary condition is now nonhomogeneous which was also necessary to consider the second-order shape derivative.

Let p ∈ (1, ∞) and let Ω ⊆ ℝN be a bounded domain with Lipschitz continuous boundary. We characterize on L2(Ω) all order-preserving semigroups that are generated by convex, lower semicontinuous, local functionals and are sandwiched between the semigroups generated by the p-Laplace operator with Dirichlet and Neumann boundary conditions. We show that every such semigroup is generated by the p-Laplace operator with Robin-type boundary conditions.

In this paper we consider the Laplace operator with Dirichlet boundary conditions on a smooth domain. We prove that it has a bounded H∞-calculus on weighted Lp-spaces for power weights which fall outside the classical class of Ap-weights. Furthermore, we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity. In particular, we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data.

Let n >= 3, Omega be a strongly Lipschitz domain of R-n and L-Omega :=-Delta + V a Schrodinger operator on L-2(Omega) with the Dirichlet boundary condition, where Delta is the Laplace operator and the nonnegative potential V belongs to the reverse Holder class RHq0(R-n) for some q(0) > n/2. Assume that the growth function phi : R-n x [0, infinity) -> [0, infinity) satisfies that phi(x, .) is an Orlicz function, phi(., t) is an element of A(infinity)(R-n) (the class of uniformly Muckenhoupt weights) and its uniformly critical lower type index i(phi) is an element of (n/n+delta, 1], where delta := min {mu(0), 2- n/q(0)} and mu(0) is an element of (0,1] denotes the critical regularity index of the heat kernels of the Laplace operator Delta on Omega. In this article, the authors first show that the heat kernels of L-Omega satisfy the Gaussian upper bound estimates and the Holder continuity. The authors then introduce the 'geometrical' Musielak-Orlicz-Hardy space H-phi,H-LRn,H-r(Omega) via H-phi,H-LRn (R-n), the Hardy space associated with L-Rn :=-Delta + Von R-n, and establish its several equivalent characterizations, respectively, in terms of the non-tangential or the vertical maximal functions or the Lusin area functions associated with L-Omega. All the results essentially improve the known results even on Hardy spaces H-LRn,r(p)(Omega) with p is an element of (n / (n + delta),1] (in this case, phi(x, t) := t(p) for all x is an element of Omega and t is an element of [0, infinity)). Copyright (C) 2016 John Wiley & Sons, Ltd.

In this paper, for degenerate diffusion equations with involution, the solvability of the direct and inverse problems for determining the right-hand side is studied. The equation with a fractional derivative in the Caputo sense is considered. The elliptic part of the studied equation involves a nonlocal analogue of the Laplace operator with a coefficient depending on the time variable. By studying these problems with respect to the time variable, we obtain a one-dimensional degenerate equation with a fractional Caputo derivative. The solution of this equation is expressed by a special function of the Kilbas-Saigo type. Similarly, for the spatial variable, we obtain a spectral problem for the nonlocal Laplace operator with the Dirichlet boundary condition. We explicitly find the eigenfunctions and eigenvalues of this problem and show the completeness of the system of eigenfunctions in space. Using the classical Fourier method, solutions to the problems under consideration are sought in the form of expansions in a series of eigenfunctions. The absolute and uniform convergence of the series, the possibility of their differentiation term by term in all variables and the absolute and uniform convergence of the differentiated series are proved. The main statements concerning the problems considered are presented in the form of existence and uniqueness theorems.

In this paper, we consider the system of [Formula: see text]-Laplacian equations with critical growth [Formula: see text] where [Formula: see text] is a bounded smooth domain in [Formula: see text] the first eigenvalue of the [Formula: see text]-Laplacian operator [Formula: see text] with the Dirichlet boundary condition, [Formula: see text] for [Formula: see text]. The existence of infinitely many sign-changing solutions is proved by the truncation method and by the concentration analysis on the approximating solutions, provided [Formula: see text].

In this article, a finding on finite element superconvergence is reported. The Laplacian operator with Dirichlet boundary condition is considered. The linear finite element solutions have an O(h2+α)(α≈0.5)‐superconvergence in l2 norm at nodes on an almost equilateral triangular mesh generated based on centroidal Voronoi tessellation, for an arbitrary 2D bounded domain. Extensive numerical examples are presented to demonstrate the superconvergence property. Copyright © 2008 John Wiley &amp; Sons, Ltd.