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Existence and surgery results
Existence of minimizers for spectral problems
One of my main research interest is the study of existence of solutions for spectral shape optimization problems in an unbounded setting. The main tool for proving existence in the bounded case is a well-known Theorem by Buttazzo and Dal Maso and it is not easy to extend it in the general case.
In this paper (with Aldo Pratelli) we show that any increasing functional of the first k eigenvalues of the Dirichlet Laplacian admits a (quasi-)open minimizer among the subsets of R^N of unit measure. In particular, there exists such a minimizer which is bounded, where the bound depends on k and N, but not on the functional.
Moreover, in the paper Boundedness of minimizers for spectral problems in R^N, I also proved that all minimizers for such a problem have diameter uniformly bounded.
The techniques developed in these works consists, roughly speaking, in taking an open set and show that if it has 'long tails' it is better to cut them (and slightly rearrange the set) so that, after rescaling to unit measure, the first k eigenvalues are decreased.
One of my main research interest is the study of existence of solutions for spectral shape optimization problems in an unbounded setting. The main tool for proving existence in the bounded case is a well-known Theorem by Buttazzo and Dal Maso and it is not easy to extend it in the general case.
In this paper (with Aldo Pratelli) we show that any increasing functional of the first k eigenvalues of the Dirichlet Laplacian admits a (quasi-)open minimizer among the subsets of R^N of unit measure. In particular, there exists such a minimizer which is bounded, where the bound depends on k and N, but not on the functional.
Moreover, in the paper Boundedness of minimizers for spectral problems in R^N, I also proved that all minimizers for such a problem have diameter uniformly bounded.
The techniques developed in these works consists, roughly speaking, in taking an open set and show that if it has 'long tails' it is better to cut them (and slightly rearrange the set) so that, after rescaling to unit measure, the first k eigenvalues are decreased.
This main idea can be slightly improved, as we did (with Dorin Bucur) in the paper A surgery result for the spectrum of the Dirichlet-Laplacian.
More precisely, we give a method to geometrically modify an open set such that the first k eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy and on a new version of Alt-Caffarelli ideas for regularity of free boundaries.
For a survey on these topics, see Recent existence results for spectral problems.
This techniques have been exploited also in a more recent paper (and in a work in progress) with B. Ruffini, see A spectral optimization problem with a nonlocal competing term.
More precisely, we give a method to geometrically modify an open set such that the first k eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy and on a new version of Alt-Caffarelli ideas for regularity of free boundaries.
For a survey on these topics, see Recent existence results for spectral problems.
This techniques have been exploited also in a more recent paper (and in a work in progress) with B. Ruffini, see A spectral optimization problem with a nonlocal competing term.
Existence of minimizers for nonlinear eigenvalues associated to capacitary measures
With Marco Degiovanni, in the paper Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.
It would be very interesting to be able to prove a surgery argument also in this setting (and thus existence of minimizers in $\mathbb{R}^N$), but the nonlinearity in the operator makes the problem much harder.
With Marco Degiovanni, in the paper Optimization results for the higher eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures we prove the existence of an optimal set for the minimization of the $k$-th variational eigenvalue of the $p$-Laplacian among $p$-quasi open sets of fixed measure included in a box of finite measure. An analogous existence result is obtained for eigenvalues of the $p$-Laplacian associated with Schr\"odinger potentials. In order to deal with these nonlinear shape optimization problems, we develop a general approach which allows to treat the continuous dependence of the eigenvalues of the $p$-Laplacian associated with sign-changing capacitary measures under $\gamma$-convergence.
It would be very interesting to be able to prove a surgery argument also in this setting (and thus existence of minimizers in $\mathbb{R}^N$), but the nonlinearity in the operator makes the problem much harder.
Regularity results
Lipschitz regularity of the eigenfunctions on optimal domains
The above mentioned existence results provide an optimal set in the class of 'quasi-open' sets. This is a very big class, hence a natural question is whether optimal set are more regular (at least open).
In this work with Dorin Bucur, Aldo Pratelli and Bozhidar Velichkov, we study the optimal sets Ω⊂R^N for spectral functionals F(λ_1(A),…,λ_p(A)), which are bi-Lipschitz with respect to each of the eigenvalues λ_1(A),…,λ_p(A) of the Dirichlet Laplacian on A, a prototype being the problem
min{λ_1(A)+⋯+λ_p(A) : A⊂R^N, |A|=1}.
We prove the Lipschitz regularity of the eigenfunctions u_1,…,u_p of the Dirichlet Laplacian on the optimal set Ω and, as a corollary, we deduce that Ω is open.
For functionals depending only on a generic subset of the spectrum, as for example λ_k(A), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.
The techniques used are based on a free boundary approach (employing ideas from Alt-Caffarelli), combined with the careful study of shape supersolutions, i.e. sets which are optimal with respect to external perturbations.
$C^(\infty)$ regularity for optimal sets
In the paper Regularity of the optimal sets for some spectral functionals with S. Terracini and B. Velichkov,we investigate the regularity issue for the functional λ_1+⋯+λ_p.
We prove an analogous result to the theorems proved by Alt, Caffarelli and Weiss for the regularity of the one-phase free boundary, that is, showing that every optimal set Ω has topological boundary made by a regular part which is $C^{\infty}$ regular, and a 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.
The main difference is that we have a vectorial setting, so we have to deal with functions that do not have constant sign and to treat the optimality condition in an appropriate way.
The main steps of our proof are the following:
In another work with S. Terracini and B. Velichkov, Regularity of the free boundary for the vectorial Bernoulli problem, we study the regularity for the free bondary of a vector-valued Bernoulli problem with no sign assumption on any of the components. This is strongly related to the regularity of optimal shapes for the k-th eigenvlaue.
We can prove that the free boundary is made by a regular part, which is locally the graph of a smooth function, a singular part which is of lower Hausdorff dimension and a set of branching points, which has finite d-1 Hausdorff measure.
For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem, to which we can apply previous results.
The main issue that we are now approaching is to prove that the set of branching points is contained in a d-2 dimensional regular submanifold. This is a very hard topic and we could obtain a result (employing a recent fundamental result by De Philippis, Spolaor and Velichkov) only for the second eigenvalue of the Dirichlet Laplacian in a box, with B. Trey and B. Velichkov: Regularity of the optimal sets for the second Dirichlet eigenvalue
Another shape optimization problem that we study with a focus on regularity of the free boundary, with G. Buttazzo, F. Maiale, G. Tortone and B. Velichkov, is related to integral functionals: Regularity of the optimal sets for a class of integral shape functionals
The energy of a domain $\Omega$ is obtained as the integral of a cost function $j(u,x)$ depending on the solution $u$ of a certain PDE problem on $\Omega$. The main feature of these functionals is that the minimality of a domain $\Omega$ cannot be translated into a variational problem for a single (real or vector valued) state function.
Our focus is the case of affine cost functions $ j(u,x)=−g(x)u+Q(x)$, where $u$ is the solution of the PDE $−\Delta u=f$ with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal $u$ from the inwards-outwards optimality of $\Omega$ and then we use the stability of $\Omega$ with respect to variations with smooth vector fields in order to study the blow-up limits of the state function $u$. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose the free boundary into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of $\partial\Omega$ we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove smoothness of the regular part of the free boundary when the data are smooth.
In a more recent paper, we also deal with the case of measure constraint, generalizing results of Aguilera Caffarelli and Spruck.
The above mentioned existence results provide an optimal set in the class of 'quasi-open' sets. This is a very big class, hence a natural question is whether optimal set are more regular (at least open).
In this work with Dorin Bucur, Aldo Pratelli and Bozhidar Velichkov, we study the optimal sets Ω⊂R^N for spectral functionals F(λ_1(A),…,λ_p(A)), which are bi-Lipschitz with respect to each of the eigenvalues λ_1(A),…,λ_p(A) of the Dirichlet Laplacian on A, a prototype being the problem
min{λ_1(A)+⋯+λ_p(A) : A⊂R^N, |A|=1}.
We prove the Lipschitz regularity of the eigenfunctions u_1,…,u_p of the Dirichlet Laplacian on the optimal set Ω and, as a corollary, we deduce that Ω is open.
For functionals depending only on a generic subset of the spectrum, as for example λ_k(A), our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.
The techniques used are based on a free boundary approach (employing ideas from Alt-Caffarelli), combined with the careful study of shape supersolutions, i.e. sets which are optimal with respect to external perturbations.
$C^(\infty)$ regularity for optimal sets
In the paper Regularity of the optimal sets for some spectral functionals with S. Terracini and B. Velichkov,we investigate the regularity issue for the functional λ_1+⋯+λ_p.
We prove an analogous result to the theorems proved by Alt, Caffarelli and Weiss for the regularity of the one-phase free boundary, that is, showing that every optimal set Ω has topological boundary made by a regular part which is $C^{\infty}$ regular, and a 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.
The main difference is that we have a vectorial setting, so we have to deal with functions that do not have constant sign and to treat the optimality condition in an appropriate way.
The main steps of our proof are the following:
- We prove a nondegeneracy lemma for the vector $U=(u_1,\dots, u_k)$ and then show that actually if $U$ is nondegenerate, than $u_1$ is nondegenerate, too. Moreover we give some density estimates.
- We give a monotonicity formula (in the spirit of Weiss) and make a blow-up analysis, showing that the blow-up limits are optimal and $1-$homogeneous.
- We study the optimality condition in the viscosity sense, we identify the regular and singular part of the topological boundary and then we reduce ourselves to a problem with only one positive function and apply the regularity result for the classical Alt-Caffarelli free boundary problem.
At the end we provide the estimates on the Hausdorff dimension for the singular part of the boundary.
In another work with S. Terracini and B. Velichkov, Regularity of the free boundary for the vectorial Bernoulli problem, we study the regularity for the free bondary of a vector-valued Bernoulli problem with no sign assumption on any of the components. This is strongly related to the regularity of optimal shapes for the k-th eigenvlaue.
We can prove that the free boundary is made by a regular part, which is locally the graph of a smooth function, a singular part which is of lower Hausdorff dimension and a set of branching points, which has finite d-1 Hausdorff measure.
For this purpose we shall exploit the NTA property of the regular part to reduce ourselves to a scalar one-phase Bernoulli problem, to which we can apply previous results.
The main issue that we are now approaching is to prove that the set of branching points is contained in a d-2 dimensional regular submanifold. This is a very hard topic and we could obtain a result (employing a recent fundamental result by De Philippis, Spolaor and Velichkov) only for the second eigenvalue of the Dirichlet Laplacian in a box, with B. Trey and B. Velichkov: Regularity of the optimal sets for the second Dirichlet eigenvalue
Another shape optimization problem that we study with a focus on regularity of the free boundary, with G. Buttazzo, F. Maiale, G. Tortone and B. Velichkov, is related to integral functionals: Regularity of the optimal sets for a class of integral shape functionals
The energy of a domain $\Omega$ is obtained as the integral of a cost function $j(u,x)$ depending on the solution $u$ of a certain PDE problem on $\Omega$. The main feature of these functionals is that the minimality of a domain $\Omega$ cannot be translated into a variational problem for a single (real or vector valued) state function.
Our focus is the case of affine cost functions $ j(u,x)=−g(x)u+Q(x)$, where $u$ is the solution of the PDE $−\Delta u=f$ with Dirichlet boundary conditions. We obtain the Lipschitz continuity and the non-degeneracy of the optimal $u$ from the inwards-outwards optimality of $\Omega$ and then we use the stability of $\Omega$ with respect to variations with smooth vector fields in order to study the blow-up limits of the state function $u$. By performing a triple consecutive blow-up, we prove the existence of blow-up sequences converging to homogeneous stable solution of the one-phase Bernoulli problem and according to the blow-up limits, we decompose the free boundary into a singular and a regular part. In order to estimate the Hausdorff dimension of the singular set of $\partial\Omega$ we give a new formulation of the notion of stability for the one-phase problem, which is preserved under blow-up limits and allows to develop a dimension reduction principle. Finally, by combining a higher order Boundary Harnack principle and a viscosity approach, we prove smoothness of the regular part of the free boundary when the data are smooth.
In a more recent paper, we also deal with the case of measure constraint, generalizing results of Aguilera Caffarelli and Spruck.
Geometric properties of optimal sets
In the paper (with Mette Iversen) Minimising convex combinations of low eigenvalues, we studied whether optimal sets for convex combinations of the first three eigenvalues are connected.
Moreover it is interesting to study geometric properties of optimal sets for functionals involving the first two eigenvalues: convexity, regularity...
With Davide Zucco (Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets) we used a new technique for proving non-convexity of minimizers for convex combinations of the two lowest eigenvalues. The main point is to employ quantitative inequalities for the second Dirichlet eigenvalue and the study of the optimal convex set for the Fraenkel 2-asymmetry, which turns out to be a "mobile-like" set.
Moreover in a paper with Aldo Pratelli (Some estimates for the higher eigenvalues of sets close to the ball) we prove quantitative estimates for the higher eigenvalues of sets with λ_1 not eccessively large, of the following form:
$|λ_k(\Omega)-\la_k(B)|\leq C(λ_1(\Omega)-\la_1(B))^α$.
Moreover it is interesting to study geometric properties of optimal sets for functionals involving the first two eigenvalues: convexity, regularity...
With Davide Zucco (Convex combinations of low eigenvalues, Fraenkel asymmetries and attainable sets) we used a new technique for proving non-convexity of minimizers for convex combinations of the two lowest eigenvalues. The main point is to employ quantitative inequalities for the second Dirichlet eigenvalue and the study of the optimal convex set for the Fraenkel 2-asymmetry, which turns out to be a "mobile-like" set.
Moreover in a paper with Aldo Pratelli (Some estimates for the higher eigenvalues of sets close to the ball) we prove quantitative estimates for the higher eigenvalues of sets with λ_1 not eccessively large, of the following form:
$|λ_k(\Omega)-\la_k(B)|\leq C(λ_1(\Omega)-\la_1(B))^α$.
Shape optimization from population dynamics
In the paper asymptotic optimal shapes in some spectral optimization problems, with B. Pellacci and G. Verzini, we study the optimization of the positive principal eigenvalue of an indefinite weighted problem, associated with the Neumann Laplacian in a box $\Omega\subset\mathbb{R}^N$, which arises in the investigation of the survival threshold in population dynamics.
When trying to minimize such eigenvalue with respect to the weight, one is lead to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures).
We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Using $\alpha$-symmetrization techniques on cones, we prove that, for suitable choices
of the box $\Omega$, the optimal shapes for this second problem are indeed spherical.
Moreover, for general $\Omega$, we show that small volume spectral drops are asymptotically spherical, with center at points of $\partial\Omega$ having large mean curvature.
In a sequent paper, Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems we study the asymptotic as the measure of the favorable set vanishes. In this case we still find a limit problem where optimal shapes are spherical and we can also prove that, if the measure of the favorable set is small enough, then it must be connected and its boundary must intersect the boundary of the box. Moreover, in a forthcoming paper, we will show that (in this asymptotic limit) the favorable region tends to be spherical and concentrate at points of higher mean curvature of the boundary of the box (if this is regular enough).
When trying to minimize such eigenvalue with respect to the weight, one is lead to consider a shape optimization problem, which is known to admit no spherical optimal shapes (despite some previously stated conjectures).
We investigate whether spherical shapes can be recovered in some singular perturbation limit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limit the so called spectral drop problem, which involves the minimization of the first eigenvalue of the mixed Dirichlet-Neumann Laplacian. Using $\alpha$-symmetrization techniques on cones, we prove that, for suitable choices
of the box $\Omega$, the optimal shapes for this second problem are indeed spherical.
Moreover, for general $\Omega$, we show that small volume spectral drops are asymptotically spherical, with center at points of $\partial\Omega$ having large mean curvature.
In a sequent paper, Singular analysis of the optimizers of the principal eigenvalue in indefinite weighted Neumann problems we study the asymptotic as the measure of the favorable set vanishes. In this case we still find a limit problem where optimal shapes are spherical and we can also prove that, if the measure of the favorable set is small enough, then it must be connected and its boundary must intersect the boundary of the box. Moreover, in a forthcoming paper, we will show that (in this asymptotic limit) the favorable region tends to be spherical and concentrate at points of higher mean curvature of the boundary of the box (if this is regular enough).
Optimization of functionals arising from physics with a local/nonlocal competiton
In the paper A spectral optimization problem with a nonlocal competing term, in collaboration with B. Ruffini, we study the minimization, under measure constraint of the functional made of the first eigenvalue of the Dirichlet Laplacian plus a Riesz repulsive potential. We prove existence of a minimizer and that only balls are minimal when the Riesz term is small, while that there are no minimizers (in suitable classes) if the Riesz term becomes dominant. We are currently studying the same kind of problem for functionals arising from the reduced Hartree equation, and several generalizations, with a focus also on nonexistence of minimizers.
In the paper A spectral optimization problem with a nonlocal competing term, in collaboration with B. Ruffini, we study the minimization, under measure constraint of the functional made of the first eigenvalue of the Dirichlet Laplacian plus a Riesz repulsive potential. We prove existence of a minimizer and that only balls are minimal when the Riesz term is small, while that there are no minimizers (in suitable classes) if the Riesz term becomes dominant. We are currently studying the same kind of problem for functionals arising from the reduced Hartree equation, and several generalizations, with a focus also on nonexistence of minimizers.
Gradient flows for eigenvalues of $L^2$ potentials:
In the paper with G. Savare' L2-Gradient Flows of Spectral Functionals we study the $L^2$-gradient flow of functionals depending on the eigenvalues of Schrodinger potentials for a wide class of differential operators associated to closed, symmetric, and coercive bilinear forms, including the case of all the Dirichlet forms (such as for second order elliptic operators in Euclidean domains or Riemannian manifolds). We are able in this setting to show the convergence of the Minimizing Movements method to an explicit generalized gradient flow equation.
In the paper with G. Savare' L2-Gradient Flows of Spectral Functionals we study the $L^2$-gradient flow of functionals depending on the eigenvalues of Schrodinger potentials for a wide class of differential operators associated to closed, symmetric, and coercive bilinear forms, including the case of all the Dirichlet forms (such as for second order elliptic operators in Euclidean domains or Riemannian manifolds). We are able in this setting to show the convergence of the Minimizing Movements method to an explicit generalized gradient flow equation.
Some ongoing stuff
- Optimization of funtionals related to the Hartree equation with a local/nonlocal competition
- Regularity results for nonvariational free boundary problems
- Nonexistence results for the Gamow problem
- Blaschke Santalo diagrams for geometric quantities
- Gradient flows in shape optimization (mainly for eigenvalues of potentials)
- Shape optimization problems for eigenvalues of the p- Laplacian
- Properties of minimizers of principal eigenvalue with indefinite weight and Neumann or Robin boundary conditions
