Existence and Regularity of Minimizers for Some Spectral Functionals with Perimeter Constraint

In this paper we prove that the shape optimization problem

has a solution for any \(k\in \mathbb{N}\) 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 \(f(\lambda_{k_{1}}(\varOmega),\dots,\lambda_{k_{p}}(\varOmega))\), for increasing functions f satisfying some suitable bi-Lipschitz type condition.

1. We recall that, thanks to the Poincar e inequality, \(\tilde{H}^{1}_{0}(\varOmega)\) is an Hilbert space with the scalar product given by

   $$\langle u,v\rangle_{\tilde{H}_0^1(\varOmega)} =\int\nabla u\cdot \nabla v\,dx. $$

2. We recall that if Ω=A∪B with \(\operatorname{dist}(A,B)>0\), \(\tilde{H}_{0}^{1}(\varOmega)=\tilde{H}^{1}_{0}(A)\oplus\tilde{H}_{0}^{1}(B)\) hence the spectrum of the Dirichlet Laplacian of Ω is given by the union of the spectrum of the Dirichlet Laplacian of A and of B.

3. A property \(\mathcal{P}\) is said to hold quasi-everywhere if

   $$\mathrm{cap}\bigl(\{\mathcal{P} \text{ is false}\}\bigr)=0. $$

4. Another way to conclude is to notice that for \(\tilde{\varOmega}\) the origin is not a regular point, a contradiction with Theorem 5.6.

5. We denote with ∂ ^M Ω the essential boundary of Ω, i.e. the complement to the set of density 1 points of Ω and of Ω ^c.

6. This can be easily seen, since any tangent cone at these points is contained in an half-space and hence it has to coincide with it, see [33, Theorem 36.5].

7. To see this, just notice that \(w_{U_{n}}\) is the solution of the Euler-Lagrange equation associated to the functional \(F:\tilde{H}^{1}_{0}(U_{n})\to \mathbb{R}\) defined by

   $$F(v)=\frac{1}{2} \int\bigl|\nabla(w_{\varOmega_n}-v)\bigr|^2\,dx=\frac{1}{2} \int | \nabla w _{\varOmega_n}|^2\,dx-\int v\,dx+\frac{1}{2} \int|\nabla v |^2\,dx, $$

   where, in the last equality, we have taken into account the equation satisfied by \(w_{\varOmega_{n}}\).

Authors and Affiliations

1. Hausdorff Center for Mathematics, Endenicher Allee 62, 53115, Bonn, Germany

   Guido De Philippis

2. Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126, Pisa, Italy

   Bozhidar Velichkov

Corresponding author

Correspondence to Bozhidar Velichkov.

Appendix: Proof of Theorem 3.1

We apply the concentration compactness principle from [29] to the sequence of characteristics functions . First notice that, being all the sets of finite measure, the isoperimetric inequality and the uniform bound on the perimeters ensure that

Let us assume that

In this case, up to subsequences, we can assume that

exists and that l∈(0,+∞). Thanks to this can rescale all our sets in such a way that |Ω _n |=1 still maintaining a uniform bound on the perimeters.

As in [29] we have that, up to subsequences, the family of concentration functions \(Q_{n}:\mathbb{R}^{+}\to \mathbb{R}^{+}\), defined by

is pointwise converging to some monotone increasing function \(Q:\mathbb{R}^{+}\to \mathbb{R}^{+}\). We now consider three different cases:

1. (i)

   lim _r→∞ Q(r)=1. In this case, up to substitute Ω _n with Ω _n +x _n for suitable \(x_{n}\in \mathbb{R}^{d}\), we have that for every ε>0 there is some R>0 such that

   $$\sup_{n\in \mathbb{N}}|\varOmega_n\setminus B_R| \le {\varepsilon}. $$

   Since the functions \(w_{\varOmega_{n}}\) are uniformly bounded in \(L^{\infty}(\mathbb{R}^{d})\) we infer that for every ε>0 there is some R>0 such that

   $$\sup_{n\in \mathbb{N}} \int_{B_R^c}w_{\varOmega_n}\,dx \le {\varepsilon}. $$

   By the compact inclusion \(BV(\mathbb{R}^{d})\hookrightarrow L^{1}_{\mathrm{loc}}(\mathbb{R}^{d})\) and \(H^{1}(\mathbb{R}^{d})\hookrightarrow L^{2}_{\mathrm{loc}}(\mathbb{R}^{d})\), we see that (up to subsequences) there are a set \(\varOmega\subset \mathbb{R}^{d}\) of unit measure such that in \(L^{1}(\mathbb{R}^{d})\) and a function \(w\in H^{1}(\mathbb{R}^{d})\) such that \(w_{\varOmega_{n}}\to w_{\varOmega}\) in \(L^{2}(\mathbb{R}^{d})\). Moreover, w≥0 on \(\mathbb{R}^{d}\) and {w>0}⊂Ω. By Proposition 2.4 and the inequality R _{w>0}≤R _Ω , we conclude that the compactness (i) holds.

2. (ii)

   lim _r→∞ Q(r)=α∈(0,1). Let ε>0, then there exits r _ε ≥1/ε such that for every R≥r _ε we have α−ε≤Q(R)≤α. By the monotonicity of Q _n (r) and the pointwise convergence to Q(r) we can find R _ε ≥r _ε +1/ε and N _ε such that

   $$\alpha-2{\varepsilon}\le Q_n(R)\le\alpha+{\varepsilon},\quad \text{for every}\ n\ge N_{\varepsilon}\ \text{and every}\ r_{\varepsilon}\le R\le R_{\varepsilon}. $$

   By the definition of Q _n the above inequality implies that there is a sequence \(x_{n}\in \mathbb{R}^{d}\) such that

   $$\alpha-3{\varepsilon}\le\bigl|\varOmega_n\cap B_R(x_n)\bigr| \le\alpha+{\varepsilon}\quad \text{for every}\ n\ge N_{\varepsilon}\ \text{and every}\ r_{\varepsilon}\le R\le R_{\varepsilon}. $$

   Defining

   $$A_n^{\varepsilon}=\varOmega_n\cap B_{r_{\varepsilon}}(x_n) \quad\text{and}\quad B_n^{\varepsilon}=\varOmega_n \setminus\overline{B}_{R_{\varepsilon}}(x_n), $$

   we see that, thanks to the choice of R _ε ,

   $$ \begin{gathered} d\bigl(A_n^{\varepsilon},B_n^{\varepsilon}\bigr)\ge R_{\varepsilon}-r_{\varepsilon}\ge1/{\varepsilon}, \\ \bigl| |A_n|-\alpha\bigr|+ \bigl| |B_n|-(1-\alpha) \bigr|\le8{\varepsilon}\quad \text{and}\quad \bigl|\varOmega_n\setminus\bigl(A_n^{\varepsilon}\cup B_n^{\varepsilon}\bigr)\bigr|\le4{\varepsilon}. \end{gathered} $$

   (6.2)

   Up to substitute r _ε and R _ε with some \(\tilde{r}_{{\varepsilon}}\in (r_{{\varepsilon}},r_{{\varepsilon}}+\sqrt{{\varepsilon}})\) and \(\tilde{R}_{{\varepsilon}}\in(R_{{\varepsilon}}-\sqrt{{\varepsilon}},R_{{\varepsilon}})\), we may suppose that

   $$\mathcal{H}^{d-1}\bigl(\partial B_{r_{\varepsilon}}(x_n)\cap \varOmega_n\bigr)+\mathcal {H}^{d-1}\bigl(\partial B_{R_{\varepsilon}}(x_n)\cap\varOmega_n\bigr)\le2\sqrt{ {\varepsilon}}, $$

   and, as a consequence,

   $$ P\bigl(A_n^{\varepsilon}\cup B_n^{\varepsilon}\bigr)\le P(\varOmega_n)+2\sqrt{{\varepsilon}}. $$

   (6.3)

   It remains to estimate the difference \(w_{\varOmega_{n}}-w_{U_{n}}\), where \(U_{n}:=A_{n}^{{\varepsilon}}\cup B_{n}^{{\varepsilon}}\). Let \(\phi\in C^{\infty}_{c}(\mathbb{R}^{d})\) be positive with support in B ₂ and equal to 1 on B ₁. For r>0, consider the function ϕ _r (x)=ϕ(x/r). Defining

   $$u_n^1:=\phi_{r_{\varepsilon}/2}(\cdot-x_n)w_{\varOmega_n} \in\tilde{H}^1_0\bigl(A_n^{\varepsilon}\bigr),\qquad u_n^2:= \bigl(1-\phi_{R_{\varepsilon}}( \cdot-x_n) \bigr)w_{\varOmega_n}\in\tilde{H}^1_0 \bigl(B_n^{\varepsilon}\bigr), $$

   we have that

   $$ \bigl\|w_{\varOmega_n}-u_n^1-u_n^2 \bigr\|_{L^2}\le4{\varepsilon}\|w_{\varOmega_n}\|_\infty, $$

   (6.4)

   where we noticed that we may choose r _ε and R _ε still satisfying all the previous assumptions and such that

   $$\bigl|\bigl(\varOmega_n\cap B_{2R_{\varepsilon}}(x_n)\bigr) \setminus B_{r_{\varepsilon}/2}(x_n)\bigr|\le4{\varepsilon}. $$

   Moreover, there is some universal constant C>0 such that

   $$ \int_{\mathbb{R}^d} \bigl|\nabla u_n^1\bigr|^2 \,dx+\int_{\mathbb{R}^d}\bigl|\nabla u_n^2\bigr|^2 \,dx- \int_{\mathbb{R}^d}|\nabla w_{\varOmega_n}|^2\,dx\le \frac{C}{r_{\varepsilon}}\le C{\varepsilon}, $$

   (6.5)

   where the last inequality follows by the choice of r _ε we made at the beginning of the proof.

   Since U _n ⊂Ω _n , we have that \(w_{U_{n}}\) is the orthogonal projection of \(w_{\varOmega_{n}}\) on the space \(\tilde{H}^{1}_{0}(U_{n})\) with respect to the \(H^{1}_{0}\) scalar product.^{Footnote 7} Hence

   $$ \begin{aligned}[b] &\int\bigl|\nabla(w_{\varOmega_n}-w_{U_n})\bigr|^2 \,dx\\ &\quad \le\int\bigl|\nabla\bigl(w_{\varOmega _n}-u_n^1-u_n^2 \bigr)\bigr|^2\,dx \\ &\quad =\int|\nabla w_{\varOmega_n}|^2\,dx-2\int\bigl(u_n^1+u_n^2 \bigr)\,dx+\int \bigl|\nabla u_n^1\bigr|^2\,dx+\int\bigl| \nabla u_n^2\bigr|^2\,dx \\ &\quad =2\int \bigl(w_{\varOmega_n}-u_n^1-u_n^2 \bigr)\,dx+\int \bigl(\bigl|\nabla u_n^1\bigr|^2+\bigl| \nabla u_n^2\bigr|^2-|\nabla w_{\varOmega_n}|^2 \bigr)\,dx\le C{\varepsilon}, \end{aligned} $$

   (6.6)

   where in the first and second equality we have taken into account the equation satisfied by \(w_{\varOmega_{n}}\) while in the last inequality we have used (6.4) and (6.5). Sending ε→0 we see that (6.2) gives points (b) and (c) of the dichotomy case statement, (6.3) gives point (d), and (6.6) together with Proposition 2.5 gives (e). Since (a) is trivially true we obtain that in this case the dichotomy (ii) holds.

3. (iii)

   lim _r→∞ Q(r)=0. In this case the first part of the statement of the vanishing case is clear, while for the second one we need some further considerations, similar to those in [6, Proposition 3.5], based on the Lieb’s Lemma (see [28]). First notice that by a truncation argument, it is enough to prove the statement in the case when all Ω _n are bounded sets. Since \(\|R_{\varOmega _{n}}\|_{\mathcal{L}(L^{2}(\mathbb{R}^{d}))}=\lambda_{1}(\varOmega_{n})^{-1}\) it is enough to prove that

   $$ \lim_{n\to\infty} \lambda_1( \varOmega_n)=+\infty. $$

   (6.7)

   Let ε>0 be fixed, R>0 be large enough and \(N\in \mathbb{N}\) be such that for all n≥N and all \(x\in \mathbb{R}\), we have |Ω _n ∩B _R (x)|≤ε. By the Lieb’s Lemma, we have that there is some \(x\in \mathbb{R}^{d}\) such that

   $$\lambda_1\bigl(\varOmega_n\cap B_R(x)\bigr) \le\lambda_1(\varOmega_n)+\lambda_1 \bigl(B_R(x)\bigr). $$

   Using that λ ₁(B _r )=r ⁻² λ ₁(B ₁), for any r>0, and that the ball minimizes λ ₁ among all sets of given measure, we have

   $$ \lambda_1(B_1){\varepsilon}^{-2/d}-R^{-2} \lambda_1(B_1)\le\lambda_1( \varOmega_n). $$

   (6.8)

   Since the left-hand side of (6.8) goes to infinity as ε→0, we obtain (6.7).

Let us assume now that (6.1) does not hold. In this case, clearly in \(L^{1}(\mathbb{R}^{d})\), hence the same arguments of case (iii) imply that we are in the vanishing case.

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