A note on the monotonicity formula of Caffarelli-Jerison-Kenig

The aim of this note is to prove the monotonicity formula of Caffarelli-Jerison-Kenig for functions, which are not necessarily continuous. We also give a detailed proof of the multiphase version of the monotonicity formula in any dimension

An Epiperimetric Inequality for the Regularity of Some Free Boundary Problems: The 2-Dimensional Case

Using a direct approach, we prove a two-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension 2, the C1,α regularity of the free boundary in the scalar one-phase and double-phase problems, and of the reduced free boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore, we show that in the vectorial case each connected component of {|u|=0} might have cusps, but they must be a finite number. © 2018 Wiley Periodicals, Inc

Multiphase shape optimization problems

This paper is devoted to the analysis of multiphase shape optimization problems, which can formally be written as min (Formula presented.) where D ⊆ ℝd is a given bounded open set, |Ωi| is the Lebesgue measure of Ωi, and m is a positive constant. For a large class of such functionals, we analyze qualitative properties of the cells and the interaction between them. Each cell is itself a subsolution for a (single-phase) shape optimization problem, from which we deduce properties like finite perimeter, inner density, separation by open sets, absence of triple junction points, etc. As main examples we consider functionals involving the eigenvalues of the Dirichlet Laplacian of each cell, i.e., Fi = λki

On the logarithmic epiperimetric inequality for the obstacle problem

We give three different proofs of the log-epiperimetric inequality at singular points for the obstacle problem. In the first, direct proof, we write the competitor explicitly; the second proof is also constructive, but this time the competitor is given through the solution of an evolution problem on the sphere. We compare the competitors obtained in the different proofs and their relation to other similar results that appeared recently. Finally, in the appendix, we give a general theorem, which can be applied also in other contexts and in which the construction of the competitor is reduced to finding a flow satisfying two differential inequalities

Existence and regularity of optimal shapes for elliptic operators with drift

This paper is dedicated to the study of shape optimization problems for the first eigenvalue of the elliptic operator with drift L= - Δ + V(x) · ∇ with Dirichlet boundary conditions, where V is a bounded vector field. In the first instance, we prove the existence of a principal eigenvalue λ1(Ω , V) for a bounded quasi-open set Ω which enjoys similar properties to the case of open sets. Then, given m> 0 and τ≥ 0 , we show that the minimum of the following non-variational problem min{λ1(Ω,V):Ω⊂Dquasi-open,|Ω|≤m,‖V‖L∞≤τ}.is achieved, where the box D⊂ Rd is a bounded open set. The existence when V is fixed, as well as when V varies among all the vector fields which are the gradient of a Lipschitz function, are also proved. The second interest and main result of this paper is the regularity of the optimal shape Ω ∗ solving the minimization problem min{λ1(Ω,∇Φ):Ω⊂Dquasi-open,|Ω|≤m},where Φ is a given Lipschitz function on D. We prove that the optimal set Ω ∗ is open and that its topological boundary ∂Ω ∗ is composed of a regular part, which is locally the graph of a C1,α function, and a singular part, which is empty if d< d∗, discrete if d= d∗ and of locally finite Hd-d∗ Hausdorff measure if d> d∗, where d∗∈ { 5 , 6 , 7 } is the smallest dimension at which there exists a global solution to the one-phase free boundary problem with singularities. Moreover, if D is smooth, we prove that, for each x∈ ∂Ω ∗∩ ∂D, ∂Ω ∗ is C1 , 1 / 2 in a neighborhood of x

Free boundary regularity for a multiphase shape optimization problem

In this paper we prove a C1,α regularity result in dimension two for almost-minimizers of the constrained one-phase Alt-Caffarelli and the two-phase Alt-Caffarelli-Friedman functionals for an energy with variable coefficients. As a consequence, we deduce the complete regularity of solutions of a multiphase shape optimization problem for the first eigenvalue of the Dirichlet Laplacian, up to the boundary of a fixed domain that acts as a geometric inclusion constraint. One of the main ingredients is a new application of the (one-phase) epiperimetric inequality up to the boundary of the constraint. While the framework that leads to this application is valid in every dimension, the epiperimetric inequality is known only in dimension two, thus the restriction on the dimension

Existence and regularity of minimizers for some spectral functionals with perimeter constraint

In this paper we prove that the shape optimization problem {λk (Ω) : Ω ⊂ ℝd, Ω open, P(Ω) = 1, |Ω| <+ ∞- has a solution for any k ∈ ℕ and dimension d. Moreover, every solution is a bounded connected open set with boundary which is C 1,α outside a closed set of Hausdorff dimension d-8. Our results are more general and apply to spectral functionals of the form λk1 (Ω)⋯ λkp (Ω)), for increasing functions f satisfying some suitable bi-Lipschitz type condition. © 2013 Springer Science+Business Media New York

Regularity of the optimal sets for some spectral functionals

In this paper we study the regularity of the optimal sets for the shape optimization problem min{λ1(Ω)+⋯+λk(Ω) : Ω⊂Rd open, |Ω|=1}, where λ1(·) , … , λk(·) denote the eigenvalues of the Dirichlet Laplacian and | · | the d-dimensional Lebesgue measure. We prove that the topological boundary of a minimizer Ωk∗ is composed of a relatively open regular part which is locally a graph of a C∞ function and a closed singular part, which is empty if d< d∗, contains at most a finite number of isolated points if d= d∗ and has Hausdorff dimension smaller than (d- d∗) if d> d∗, where the natural number d∗∈ [ 5 , 7 ] is the smallest dimension at which minimizing one-phase free boundaries admit singularities. To achieve our goal, as an auxiliary result, we shall extend for the first time the known regularity theory for the one-phase free boundary problem to the vector-valued case

Almost everywhere uniqueness of blow-up limits for the lower dimensional obstacle problem

We answer a question left open in [4] and [3], by proving that the blow-up of minimizers u of the lower dimensional obstacle problem is unique at generic point of the free boundary

Numerical Calibration of Steiner trees

In this paper we propose a variational approach to the Steiner tree problem, which is based on calibrations in a suitable algebraic environment for polyhedral chains which represent our candidates. This approach turns out to be very efficient from numerical point of view and allows to establish whether a given Steiner tree is optimal. Several examples are provided