Let $Ω\subset\mathbb{R}^N$, $N\geq 1$, be an open bounded connected set. We consider the indefinite weighted eigenvalue problem $-Δu =λm u$ in $Ω$ with $λ\in \mathbb{R}$, $m\in L^\infty(Ω)$ and with homogeneous Neumann boundary conditions. We study weak* continuity, convexity and Gâteaux differentiability of the map $m\mapsto1/λ_1(m)$, where $λ_1(m)$ is the principal eigenvalue. Then, denoting by $\mathcal{G}(m_0)$ the class of rearrangements of a fixed weight $m_0$, under the assumptions that $m_0$ is positive on a set of positive Lebesgue measure and $\int_Ωm\,dx<0$, we prove the existence and a characterization of minimizers of $λ_1(m)$ and the non-existence of maximizers. Finally, we show that, if $Ω$ is a cylinder, then every minimizer is monotone with respect to the direction of the generatrix. In the context of the population dynamics, this kind of problems arise from the question of determining the optimal spatial location of favourable and unfavourable habitats for a population to survive.
We establish general assumptions under which a constrained vari- ational problem involving the fractional gradient and a local nonlin- earity admits minimizers.
We obtain monotonicity properties for minima and stable solutions of general energy functionals of the type $$ \int F(\nabla u, u, x) dx $$ under the assumption that a certain integral grows at most quadratically at infinity. As a consequence we obtain several rigidity results of global solutions in low dimensions.
We establish a new monotonicity formula for minimizers of the Mumford-Shah functional in planar domains. Our formula follows the spirit of Bucur-Luckhaus, but works with the David-Léger entropy instead of the energy. Interestingly, this allows for a sharp truncation threshold. In particular, our monotonicity formula is able to discriminate between points at a $C^1$ interface and any other type of singularity in terms of a finite gap in the entropy. As a corollary, we prove an optimal lower bound on the energy density around any nonsmooth point for minimizers of the Mumford-Shah functional.
Let $1<q<p$ and $a\in C(\overlineΩ)$ be sign-changing, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N}$. We show that the functional \[ I_{q}(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] has exactly one nonnegative minimizer $U_{q}$ (in $W_{0}^{1,p}(Ω)$ or $W^{1,p}(Ω)$). In addition, we prove that $U_{q}$ is the only possible \textit{positive} solution of the associated Euler-Lagrange equation, which shows that this equation has at most one positive solution. Furthermore, we show that if $q$ is close enough to $p$ then $U_{q}$ is positive, which als…
Let $Ω\subset\mathbb{R}^N$, $N\geq 2$, be an open bounded connected set. We consider the fractional weighted eigenvalue problem $(-Δ)^s u =λρu$ in $Ω$ with homogeneous Dirichlet boundary condition, where $(-Δ)^s$, $s\in (0,1)$, is the fractional Laplacian operator, $λ\in \mathbb{R}$ and $ ρ\in L^\infty(Ω)$. We study weak* continuity, convexity and Gâteaux differentiability of the map $ρ\mapsto1/λ_1(ρ)$, where $λ_1(ρ)$ is the first positive eigenvalue. Moreover, denoting by $\mathcal{G}(ρ_0)$ the class of rearrangements of $ρ_0$, we prove the existence of a minimizer of $λ_1(ρ)$ when $ρ$ varies…
An abstract linking result for Cerami sequences is proved without the Cerami condition. It is applied directly in order to prove the existence of critical points for a class of indefinite problems in infinite dimensional Hilbert Spaces. The main applications are given to Hamiltonian systems and Schrodinger equations. Here spectral properties of the operators are exploited and hypotheses of monotonicity on the nonlinearities are discarded.
We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and alternative characterizations of the first and second eigenvalues. Then, by means of such characterizations, we prove strict decreasing monotonicity of such eigenvalues with respect to the weight function.
We study global Mumford-Shah minimizers in $\R^N$, introduced by Bonnet as blow-up limits of Mumford-Shah minimizers. We prove a new monotonicity formula for the energy of $u$ when the singular set $K$ is contained in a smooth enough cone. We then use this monotonicity to prove that for any reduced global minimizer $(u,K)$ in $\R^3$, if $K$ is contained in a half-plane and touching its edge, then it is the half-plane itself. This partially answers to a question of Guy David.
In this paper, we prove the existence of minimizers of a class of multi-constrained variational problems. We consider systems involving a nonlinearity that does not satisfy compactness, monotonicity, neither symmetry properties. Our approach hinges on the concentration-compactness approach. In the second part, we will treat orthogonal constrained problems for another class of integrands using density matrices method.
We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including gradient flows, some nonlocal problems, and systems of nonlinear parabolic equations. Our method is based on the so-called Weighted-Energy-Dissipation (WED) variational approach. This consists in defining a global parameter-dependent functional over entire trajectories and proving that its minimizers converge to solutions to the target problem as the parameter go…
We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $λ> 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_λ$ and determine the monotonicity and co…
We analyze the topological structure of the Nehari set for a class of functionals depending on a real parameter $λ$, and having two degrees of homogeneity. A special attention is paid to the extremal parameter $λ^*$, which is the threshold value for the Nehari set to be given by a {\it natural constraint}. The main difficulty arises when $λ>λ^*$, as the energy functional may be unbounded from below over the Nehari set. In such situation we prove the existence of local minimizers of the functional constrained to this set. We unify and extend previous existence and multiplicity results for criti…
We consider a nonlinear elliptic equation driven by the Robin $p$-Laplacian plus an indefinite potential. In the reaction we have the competing effects of a strictly $(p-1)$-sublinear parametric term and of a $(p-1)$-linear and nonuniformly nonresonant term. We study the set of positive solutions as the parameter $λ>0$ varies. We prove a bifurcation-type result for large values of the positive parameter $λ$. Also, we show that for all admissible $λ>0$, the problem has a smallest positive solution $\overline{u}_λ$ and we study the monotonicity and continuity properties of the map $λ\mapsto\over…
We study almost minimizers for the thin obstacle problem with variable Hölder continuous coefficients and zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin space. Under an additional assumption of quasisymmetry, we establish the optimal growth of almost minimizers as well as the regularity of the regular set and a structural theorem on the singular set. The proofs are based on the generalization of Weiss- and Almgren-type monotonicity formulas for almost minimizers established earlier in the case of constant coefficients.
We study fractional parabolic equations with indefinite nonlinearities $$ \frac{\partial u} {\partial t}(x,t) +(-Δ)^s u(x,t)= x_1 u^p(x, t),\,\, (x, t) \in \mathbb{R}^n \times \mathbb{R}, $$ where $0<s<1$ and $1<p<\infty$. We first prove that all positive bounded solutions are monotone increasing along the $x_1$ direction. Based on this we derive a contradiction and hence obtain non-existence of solutions. These monotonicity and nonexistence results are crucial tools in a priori estimates and complete blow-up for fractional parabolic equations in bounded domains. To this end, we introduce seve…
We consider Anzellotti-type almost minimizers for the thin obstacle (or Signorini) problem with zero thin obstacle and establish their $C^{1,β}$ regularity on the either side of the thin manifold, the optimal growth away from the free boundary, the $C^{1,γ}$ regularity of the regular part of the free boundary, as well as a structural theorem for the singular set. The analysis of the free boundary is based on a successful adaptation of energy methods such as a one-parameter family of Weiss-type monotonicity formulas, Almgren-type frequency formula, and the epiperimetric and logarithmic epiperim…
We continue the analysis of the two-phase free boundary problems initiated in \cite{DK}, where we studied the linear growth of minimizers in a Bernoulli type free boundary problem at the non-flat points and the related regularity of free boundary. There, we also defined the functional $$ϕ_p(r,u,x_0)=\frac1{r^4}\int_{B_r(x_0)}\frac{|\nabla u^+(x)|^p}{|x-x_0|^{N-2}}dx\int_{B_r(x_0)}\frac{|\nabla u^-(x)|^p}{|x-x_0|^{N-2}}dx$$ where $x_0$ is a free boundary point, i.e. $x_0\in\partial\{u>0\}$ and $u$ is a minimizer of the functional $$J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le …
We consider $\mathbb{S}^2$-valued maps on a domain $Ω\subset\mathbb{R}^N$ minimizing a perturbation of the Dirichlet energy with vertical penalization in $Ω$ and horizontal penalization on $\partialΩ$. We first show the global minimality of universal constant configurations in a specific range of the physical parameters using a Poincaré-type inequality. Then, we prove that any energy minimizer takes its values into a fixed meridian of the sphere $\mathbb{S}^2$, and deduce uniqueness of minimizers up to the action of the appropriate symmetry group. We also prove a comparison principle for minim…
We present a new approach to solve a Schrödinger Equation autonomous at infinity, by identifying the relation between the arrangement of the spectrum of the concerned operator and the behavior of the nonlinearity at zero and at infinity. In order to apply variational methods, we set up a suitable linking structure depending on the growth of the nonlinear term and making use of information about the autonomous problem at infinity. Our method allows us to circumvent the lack of compactness. The main novelty is that none monotonicity assumption is required on the nonlinearity, which may be sign-c…
We consider a fractional version of Gauss capillarity energy. A suitable extension problem is introduced to derive a boundary monotonicity formula for local minimizers of this fractional capillarity energy. As a consequence, blow-up limits of local minimizers are shown to subsequentially converge to minimizing cones. Finally, we show that in the planar case there is only one possible fractional minimizing cone, the one determined by the fractional version of Young's law.
A two-dimensional free boundary transmission problem arising in the modeling of an electrostatically actuated plate is considered and a representation formula for the derivative of the associated electrostatic energy with respect to the deflection of the plate is derived. The latter paves the way for the construction of energy minimizers and also provides the Euler-Lagrange equation satisfied by these minimizers. A by-product is the monotonicity of the electrostatic energy with respect to the deflection.
We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up to the fixed boundary under Neumann boundary conditions. Our proof uses an almost monotonicity formula for the Alt-Caffarelli-Friedman functional restricted to the convex domain. This requires a variant of the classical Friedland-Hayman inequality for geodesically convex subsets of the sphere with Neumann boundary conditions. To apply this inequality, in addition to convexity, we require a Dini condition governing the…
We consider the problem $$ (P_λ)\quad -Δ_{p}u=λu^{p-1}+a(x)u^{q-1},\quad u\geq0\quad\mbox{ in }Ω$$ under Dirichlet or Neumann boundary conditions. Here $Ω$ is a smooth bounded domain of $\mathbb{R}^{N}$ ($N\geq1$), $λ\in\mathbb{R}$, $1<q<p$, and $a\in C(\overlineΩ)$ changes sign. These conditions enable the existence of dead core solutions for this problem, which may admit multiple nontrivial solutions. We show that for $λ<0$ the functional \[ I_λ(u):=\int_Ω\left( \frac{1}{p}|\nabla u|^{p}-\frac{λ}{p}|u|^{p}-\frac{1}{q}a(x)|u|^{q}\right) , \] defined in $X=W_{0}^{1,p}(Ω)$ or $X=W^{1,p}(Ω)$, ha…
In this work, we prove an existence result for an optimal partition problem of the form $$\min \{F_s(A_1,\dots,A_m)\colon A_i \in \mathcal{A}_s, \, A_i\cap A_j =\emptyset \mbox{ for } i\neq j\},$$ where $F_s$ is a cost functional with suitable assumptions of monotonicity and lowersemicontinuity, $\mathcal{A}_s$ is the class of admissible domains and the condition $A_i\cap A_j =\emptyset$ is understood in the sense of the Gagliardo $s$-capacity, where $0<s<1$. Examples of this type of problem are related to the fractional eigenvalues. In addition, we prove some type of convergence of the $s$-mi…
This article concerns the antisymmetry, uniqueness, and monotonicity properties of solutions to some elliptic functionals involving weights and a double well potential. In the one-dimensional case, we introduce the continuous odd rearrangement of an increasing function and we show that it decreases the energy functional when the weights satisfy a certain convexity-type hypothesis. This leads to the antisymmetry or oddness of increasing solutions (and not only of minimizers). We also prove a uniqueness result (which leads to antisymmetry) where a convexity-type condition by Berestycki and Niren…
This doctoral thesis is devoted to the analysis of some minimization problems that involve nonlocal functionals. We are mainly concerned with the $s$-fractional perimeter and its minimizers, the $s$-minimal sets. We investigate the behavior of sets having (locally) finite fractional perimeter and we establish existence and compactness results for (locally) $s$-minimal sets. We study the $s$-minimal sets in highly nonlocal regimes, that correspond to small values of the fractional parameter $s$. We introduce a functional framework for studying those $s$-minimal sets that can be globally written…
In this paper we give an unified approach to some questions arising in different fields of nonlinear analysis, namely: (a) the study of the structure of the Fuč\'ık spectrum and (b) possible variants and extensions of the monotonicity formula by Alt--Caffarelli--Friedman \cite{acf}. In the first part of the paper we present a class of optimal partition problems involving the first eigenvalue of the Laplace operator. Beside establishing the existence of the optimal partition, we develop a theory for the extremality conditions and the regularity of minimizers. As a first application of this appr…
We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $λ$ varies. We also show that there exists a minimal positive solution $\overline{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$. Special attention is given to the particular case of the $p$-Laplacian.
We present a detailed analysis of one-dimensional Néel walls in thin uniaxial ferromagnetic films in the presence of an in-plane applied external field in the direction normal to the easy axis. Within the reduced one-dimensional thin film model, we formulate a non-local variational problem whose minimizers are given by one-dimensional Néel wall profiles. We prove existence, uniqueness (up to translations and reflections), regularity, strict monotonicity and the precise asymptotics of the decay of the minimizers in the considered variational problem.
Periodic waves in the modified Korteweg-de Vries (mKdV) equation are revisited in the setting of the fractional Laplacian. Two families of solutions in the local case are given by the sign-definite dnoidal and sign-indefinite cnoidal solutions. Both solutions can be characterized in the general fractional case as global minimizers of the quadratic part of the energy functional subject to the fixed $L^4$ norm: the sign-definite (sign-indefinite) solutions are obtained in the subspace of even (odd) functions. Morse index is computed for both solutions and the spectral stability criterion is deri…
This paper provides the first variational proof of the existence of periodic nonlocal-CMC surfaces. These are nonlocal analogues of the classical Delaunay cylinders. More precisely, we show the existence of a set in $\mathbb{R}^n$ which is periodic in one direction, has a prescribed (but arbitrary) volume within a slab orthogonal to that direction, has constant nonlocal mean curvature, and minimizes an appropriate periodic version of the fractional perimeter functional under the volume constraint. We show, in addition, that the set is cylindrically symmetric and, more significantly, that it is…
We establish existence and qualitative properties of saddle-shaped solutions of the elliptic fractional equation $(-Δ)^{1/2}u=f(u)$ in all the space $\re^{2m}$, where $f$ is of bistable type. These solutions are odd with respect to the Simons cone and even with respect to each coordinate. More precisely, we prove the existence of a saddle-shaped solution in every even dimension $2m$, as well as its monotonicity properties, asymptotic behaviour, and instability in dimensions $2m=4$ and $2m=6$. These results are relevant in connection with the analog for fractional equations of a conjecture of D…
Maximization and minimization of the principal eigenvalue of the Laplacian with indefinite weight under Dirichlet and Robin boundary conditions on classes of rearrangements
本文研究带不定权重的拉普拉斯算子主特征值在重排类上的最小化与最大化问题，统一处理 Dirichlet 和 Robin 边界条件，并给出 Dirichlet 情形下唯一极大元的完整刻画。
We prove partial regularity of stationary solutions and minimizers $u$ from a set $Ω\subset \mathbb R^n$ to a Riemannian manifold $N$, for the functional $\int_ΩF(x,u,|\nabla u|^2) dx$. The integrand $F$ is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an $ε$-regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable.
In this paper we classify the nonnegative global minimizers of the functional \[ J_F(u)=\int_ΩF(|\nabla u|^2)+λ^2χ_{\{u>0\}}, \] where $F$ satisfies some structural conditions and $χ_D$ is the characteristic function of a set $D\subset \mathbb R^n$. We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\partial\{u>0\}$ can be seen as a rectifiable $n-1$ varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded and use Allard's monotonicity formula to prove the existen…
For open radial sets $Ω\subset \mathbb{R}^N$, $N\geq 2$ we consider the nonlinear problem \[ (P)\quad Iu=f(|x|,u) \quad\text{in $Ω$,}\quad u\equiv 0\quad \text{on $\mathbb{R}^N\setminus Ω$ and }\lim_{|x|\to\infty} u(x)=0, \] where $I$ is a nonlocal operator and $f$ is a nonlinearity. Under mild symmetry and monotonicity assumptions on $I$, $f$ and $Ω$ we show that any continuous bounded solution of $(P)$ is axial symmetric once it satisfies a simple reflection inequality with respect to a hyperplane. In the special case where $f$ does not depend on $|x|$, we show that any nonnegative nontrivia…
This paper is dedicated to the spectral optimization problem $$ \mathrm{min}\left\{λ_1^s(Ω)+\cdots+λ_m^s(Ω) + Λ\mathcal{L}_n(Ω)\colon Ω\subset D \mbox{ s-quasi-open}\right\} $$ where $Λ>0, D\subset \mathbb{R}^n$ is a bounded open set and $λ_i^s(Ω)$ is the $i$-th eigenvalues of the fractional Laplacian on $Ω$ with Dirichlet boundary condition on $\mathbb{R}^n\setminus Ω$. We first prove that the first $m$ eigenfunctions on an optimal set are locally Hölder continuous in the class $C^{0,s}$ and, as a consequence, that the optimal sets are open sets. Then, via a blow-up analysis based on a Weiss …
We consider a semilinear elliptic equation with Dirichlet boundary conditions in a smooth, possibly unbounded, domain. Under suitable assumptions, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means that the estimate is independent of the domain. The main advantage of our approach is that it allows us to remove a restrictive monotonicity assumption that was imposed in the recent paper. In addition, we can remove a non-degeneracy condition that was assumed in the latter reference. Furtherm…
In this paper, we investigate an optimal design problem motivated by some issues arising in population dynamics. In a nutshell, we aim at determining the optimal shape of a region occupied by resources for maximizing the survival ability of a species in a given box and we consider the general case of Robin boundary conditions on its boundary. Mathematically, this issue can be modeled with the help of an extremal indefinite weight linear eigenvalue problem. The optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to the weight, under a L 1 constra…
We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in our previous paper. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this pa…
Normalized solutions of $L^2$-supercritical NLS equations with periodic potentials
School of Mathematics East China University of Science and Technology（华东理工大学数学学院）；Department of Mathematics Faculty of Science and Technology Keio University（庆应义塾大学理工学部数学系）
