Showing papers in "Calculus of Variations and Partial Differential Equations in 2018"

Regularity for general functionals with double phase

Abstract: We prove sharp regularity results for a general class of functionals of the type $$\begin{aligned} w \mapsto \int F(x, w, Dw) \, dx, \end{aligned}$$ featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral $$\begin{aligned} w \mapsto \int b(x,w)(|Dw|^p+a(x)|Dw|^q) \, dx,\quad 1

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

Abstract: We investigate quantitative properties of nonnegative solutions $$u(x)\ge 0$$ to the semilinear diffusion equation $$\mathcal {L}u= f(u)$$ , posed in a bounded domain $$\Omega \subset \mathbb {R}^N$$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $$\mathcal {L}$$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $$(-\Delta )^s$$ ( $$0

The Allen-Cahn equation on closed manifolds

Abstract: We study global variational properties of the space of solutions to $$-\varepsilon ^2\Delta u + W'(u)=0$$ on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if $$\varepsilon $$ is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as $$\varepsilon \rightarrow 0$$ and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed $$\varepsilon $$ , as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as $$\varepsilon \rightarrow 0$$ , around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with $$\mathrm{Ric}_M>0$$ , the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.

Bakry–Émery curvature and diameter bounds on graphs

Abstract: We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Emery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet–Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672–698, 2018) and Horn et al. (J fur die reine und angewandte Mathematik (Crelle’s J), 2017, https://doi.org/10.1515/crelle-2017-0038 ) and solve a conjecture from Cushing et al. (Bakry–Emery curvature functions of graphs, 2016).

Concentration-compactness principle for Trudinger–Moser inequalities on Heisenberg groups and existence of ground state solutions

Abstract: Let $$\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}$$ be the n-dimensional Heisenberg group, $$Q=2n+2$$ be the homogeneous dimension of $$\mathbb {H}^{n}$$ . We extend the well-known concentration-compactness principle on finite domains in the Euclidean spaces of Lions (Rev Mat Iberoam 1:145–201, 1985) to the setting of the Heisenberg group $$\mathbb {H}^{n}$$ . Furthermore, we also obtain the corresponding concentration-compactness principle for the Sobolev space $${ HW}^{1,Q}(\mathbb {H}^{n}) $$ on the entire Heisenberg group $$\mathbb {H}^{n}$$ . Our results improve the sharp Trudinger–Moser inequality on domains of finite measure in $$\mathbb {H}^{n}$$ by Cohn and Lu (Indiana Univ Math J 50(4):1567–1591, 2001) and the corresponding one on the whole space $$\mathbb {H}^n$$ by Lam and Lu (Adv Math 231:3259–3287, 2012). All the proofs of the concentration-compactness principles for the Trudinger–Moser inequalities in the literature even in the Euclidean spaces use the rearrangement argument and the Polya–Szego inequality. Due to the absence of the Polya–Szego inequality on the Heisenberg group, we will develop a different argument. Our approach is surprisingly simple and general and can be easily applied to other settings where symmetrization argument does not work. As an application of the concentration-compactness principle, we establish the existence of ground state solutions for a class of Q- Laplacian subelliptic equations on $$\mathbb {H}^{n}$$ : $$\begin{aligned} -\mathrm {div}\left( \left| abla _{\mathbb {H}}u\right| ^{Q-2} abla _{\mathbb {H}}u\right) +V(\xi ) \left| u\right| ^{Q-2}u=\frac{f(u) }{\rho (\xi )^{\beta }} \end{aligned}$$ with nonlinear terms f of maximal exponential growth $$\exp (\alpha t^{\frac{Q}{Q-1}})$$ as $$t\rightarrow +\infty $$ . All the results proved in this paper hold on stratified groups with the same proofs. Our method in this paper also provide a new proof of the classical concentration-compactness principle for Trudinger-Moser inequalities in the Euclidean spaces without using the symmetrization argument.

Ground states in the diffusion-dominated regime

Abstract: We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and $$C^\infty $$ inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

Compactness analysis for free boundary minimal hypersurfaces

Abstract: We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.

Comparison and vanishing theorems for Kähler manifolds

Abstract: In this paper, we consider orthogonal Ricci curvature $$Ric^{\perp }$$ for Kahler manifolds, which is a curvature condition closely related to Ricci curvature and holomorphic sectional curvature. We prove comparison theorems and a vanishing theorem related to these curvature conditions, and construct various examples to illustrate subtle relationship among them. As a consequence of the vanishing theorem, we show that any compact Kahler manifold with positive orthogonal Ricci curvature must be projective. This result complements a recent result of Yang (RC-positivity, rational connectedness, and Yau’s conjecture. arXiv:1708.06713 ) on the projectivity under the positivity of holomorphic sectional curvature. The simply-connectedness is shown when the complex dimension is smaller than five. Further study of compact Kahler manifolds with $$Ric^{\perp }>0$$ is carried in Ni et al. (Manifolds with positive orthogonal Ricci curvature. arXiv:1806.10233 ).

Sharp conditions for the existence of boundary blow-up solutions to the Monge–Ampère equation

Abstract: In this paper we give sharp conditions on K(x) and f(u) for the existence of strictly convex solutions to the boundary blow-up Monge–Ampere problem $$\begin{aligned} M[u](x)=K(x)f(u) \quad \hbox {for } x \in \Omega ,\; u(x)\rightarrow +\,\infty \quad \hbox {as } \mathrm{dist}(x,\partial \Omega )\rightarrow 0. \end{aligned}$$ Here $$M[u]=\det \, (u_{x_{i}x_{j}})$$ is the Monge–Ampere operator, and $$\Omega $$ is a smooth, bounded, strictly convex domain in $$ \mathbb {R}^N \, (N\ge 2)$$ . Further results are obtained for the special case that $$\Omega $$ is a ball. Our approach is largely based on the construction of suitable sub- and super-solutions.

The semi-classical limit of large fermionic systems

Abstract: We study a system of N fermions in the regime where the intensity of the interaction scales as 1 / N and with an effective semi-classical parameter $$\hbar =N^{-1/d}$$ where d is the space dimension. For a large class of interaction potentials and of external electromagnetic fields, we prove the convergence to the Thomas–Fermi minimizers in the limit $$N\rightarrow \infty $$ . The limit is expressed using many-particle coherent states and Wigner functions. The method of proof is based on a fermionic de Finetti–Hewitt–Savage theorem in phase space and on a careful analysis of the possible lack of compactness at infinity.

Existence and phase separation of entire solutions to a pure critical competitive elliptic system

Abstract: We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u=\mu _{1}|u|^{{2}^{*}-2}u+\lambda \alpha |u|^{\alpha -2}|v|^{\beta }u,\\ -\,\Delta v=\mu _{2}|v|^{{2}^{*}-2}v+\lambda \beta |u|^{\alpha } |v|^{\beta {-2} }v,\\ u,v\in D^{1,2}({\mathbb {R}}^{N}), \end{array}\right. } \end{aligned}$$ where $$N\ge 4,$$ $$2^{*}:=\frac{2N}{N-2}$$ is the critical Sobolev exponent, $$\alpha ,\beta \in (1,2],$$ $$\alpha +\beta =2^{*},$$ $$\mu _{1},\mu _{2}>0,$$ and $$\lambda <0.$$ We show that these solutions exhibit phase separation as $$\lambda \rightarrow -\,\infty ,$$ and we give a precise description of their limit domains. If $$\mu _{1}=\mu _{2}$$ and $$\alpha =\beta $$ , we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.

Laplacian cut-offs, porous and fast diffusion on manifolds and other applications

Abstract: We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without any assumptions on the topology or the injectivity radius. Along the way we prove a generalization of the Li-Yau gradient estimate which is of independent interest. We then apply our cut-offs to the study of the fast and porous media diffusion, of $$L^q$$ -properties of the gradient and of the self-adjointness of Schrodinger-type operators.

Higher differentiability for solutions to a class of obstacle problems

Abstract: We establish the higher differentiability of integer and fractional order of the solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra (integer or fractional) differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form $$\begin{aligned} \int _{\Omega } \langle \mathcal {A}(x, Du), D(\varphi - u) \rangle \, dx \ge 0 \qquad \forall \varphi \in \mathcal {K}_{\psi }(\Omega ) \end{aligned}$$ where $$\mathcal {A}$$ is a p-harmonic type operator, $$\psi \in W^{1,p}(\Omega )$$ is a fixed function called obstacle and $$\mathcal {K}_{\psi }=\{w \in W^{1,p}(\Omega ): w \ge \psi \,\,$$ a.e. in $$\Omega \}$$ is the class of the admissible functions. We prove that an extra differentiability assumption on the gradient of the obstacle transfers to Du with no losses in the natural exponent of integrability, provided the partial map $$x\mapsto \mathcal {A}(x,\xi )$$ possesses a suitable differentiability property measured either in the scale of the Sobolev space $$W^{1,n}$$ or in that of the critical Besov–Lipschitz spaces $$B^\alpha _{\frac{n}{\alpha }, q}$$ , for a suitable $$1\le q\le +\infty $$ .

Maximization of the total population in a reaction–diffusion model with logistic growth

Abstract: This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction–diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang–bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (Nonlinear Anal Real World Appl 11(2):688–704, 2010). To this purpose, we compute the first and second variations of the total population. When the growth rate is not of bang–bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. When the first variation becomes zero, we prove that the second variation is positive. These results implies that the bang–bang property is essential for the maximization of total population.

The p -capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems

Abstract: In this paper, combining the p-capacity for $$p\in (1, n)$$ with the Orlicz addition of convex domains, we develop the p-capacitary Orlicz–Brunn–Minkowski theory. In particular, the Orlicz $$L_{\phi }$$ mixed p-capacity of two convex domains is introduced and its geometric interpretation is obtained by the p-capacitary Orlicz–Hadamard variational formula. The p-capacitary Orlicz–Brunn–Minkowski and Orlicz–Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The p-capacitary Orlicz–Minkowski problem is proposed and solved under some mild conditions on the involving functions and measures. In particular, we provide the solutions for the normalized p-capacitary $$L_q$$ Minkowski problems with $$q>1$$ for both discrete and general measures.

Variational approach for the Stokes and Navier–Stokes systems with nonsmooth coefficients in Lipschitz domains on compact Riemannian manifolds

Abstract: The purpose of this paper is to show well-posedness results for Dirichlet problems for the Stokes and Navier–Stokes systems with $$L^{\infty }$$ -variable coefficients in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. First, we refer to the Dirichlet problem for the nonsmooth coefficient Stokes system on Lipschitz domains in compact Riemannian manifolds and show its well-posedness by employing a variational approach that reduces the boundary value problem of Dirichlet type to a variational problem defined in terms of two bilinear continuous forms, one of them satisfying a coercivity condition and another one the inf-sup condition. We show also the equivalence between some transmission problems for the nonsmooth coefficient Stokes system in complementary Lipschitz domains on compact Riemannian manifolds and their mixed variational counterparts, and then their well-posedness in $$L^2$$ -based Sobolev spaces by using the remarkable Necas–Babuska–Brezzi technique (see Babuska in Numer Math 20:179–192, 1973; Brezzi in RAIRO Anal Numer R2:129–151, 1974; Necas in Rev Roum Math Pures Appl 9:47–69, 1964). As a consequence of these well-posedness results we define the layer potential operators for the nonsmooth coefficient Stokes system on Lipschitz surfaces in compact Riemannian manifolds, and provide their main mapping properties. These properties are used to construct explicitly the solution of the Dirichlet problem for the Stokes system. Further, we combine the well-posedness of the Dirichlet problem for the nonsmooth coefficient Stokes system with a fixed point theorem to show the existence of a weak solution to the Dirichlet problem for the nonsmooth variable coefficient Navier–Stokes system in $$L^2$$ -based Sobolev spaces in Lipschitz domains on compact Riemannian manifolds. The well developed potential theory for the smooth coefficient Stokes system on compact Riemannian manifolds (cf. Dindos and Mitrea in Arch Ration Mech Anal 174:1–47, 2004; Mitrea and Taylor in Math Ann 321:955–987, 2001) is also discussed in the context of the potential theory developed in this paper.

Geometric inequalities on Heisenberg groups

Abstract: We establish geometric inequalities in the sub-Riemannian setting of the Heisenberg group $$\mathbb H^n$$ . Our results include a natural sub-Riemannian version of the celebrated curvature-dimension condition of Lott–Villani and Sturm and also a geodesic version of the Borell–Brascamp–Lieb inequality akin to the one obtained by Cordero-Erausquin, McCann and Schmuckenschlager. The latter statement implies sub-Riemannian versions of the geodesic Prekopa–Leindler and Brunn–Minkowski inequalities. The proofs are based on optimal mass transportation and Riemannian approximation of $$\mathbb H^n$$ developed by Ambrosio and Rigot. These results refute a general point of view, according to which no geometric inequalities can be derived by optimal mass transportation on singular spaces.

Global Sobolev regularity for general elliptic equations of p-Laplacian type

Abstract: We derive global gradient estimates for $$W^{1,p}_0(\Omega )$$ -weak solutions to quasilinear elliptic equations of the form $$\begin{aligned} \mathrm {div\,}\mathbf {a}(x,u,Du)=\mathrm {div\,}(|F|^{p-2}F) \end{aligned}$$ over n-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to x and merely continuous in u. Our result highly improves the known regularity results available in the literature. Actually, we are able not only to weaken the Lipschitz continuity with respect to u of the nonlinearity to only uniform continuity, but we also find a very lower level of geometric assumption on the boundary of the domain to ensure a global character of the gradient estimates obtained.

A three-dimensional symmetry result for a phase transition equation in the genuinely nonlocal regime

Abstract: We consider bounded solutions of the nonlocal Allen–Cahn equation $$\begin{aligned} (-\Delta )^s u=u-u^3\qquad { \text{ in } }\mathbb {R}^3, \end{aligned}$$ under the monotonicity condition $$\partial _{x_3}u>0$$ and in the genuinely nonlocal regime in which $$s\in \left( 0,\frac{1}{2}\right) $$ . Under the limit assumptions $$\begin{aligned} \lim _{x_n\rightarrow -\infty } u(x',x_n)=-1\quad { \text{ and } }\quad \lim _{x_n\rightarrow +\infty } u(x',x_n)=1, \end{aligned}$$ it has been recently shown in Dipierro et al. (Improvement of flatness for nonlocal phase transitions, 2016) that u is necessarily 1D, i.e. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by De Giorgi (Proceedings of the international meeting on recent methods in nonlinear analysis (Rome, 1978), Pitagora, Bologna, pp 131–188, 1979).

Complete $$\lambda $$ λ -hypersurfaces of weighted volume-preserving mean curvature flow

Abstract: In this paper, we introduce a special class of hypersurfaces which are called $$\lambda $$ -hypersurfaces related to a weighted volume preserving mean curvature flow in the Euclidean space. We prove that $$\lambda $$ -hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete $$\lambda $$ -hypersurfaces with polynomial area growth and $$H-\lambda \ge 0$$ . The classification result generalizes the results of Huisken (J Differ Geom 31:285–299, 1990) and Colding and Minicozzi (Ann Math 175:755–833, 2012).

Generalization of unfolding operator for highly oscillating smooth boundary domains and homogenization

Abstract: Unfolding operators have been introduced and used to study homogenization problems. Initially, they were introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth. In this article, we develop new unfolding operators, where the oscillations can be smooth and hence they have wider applications. We have demonstrated by developing unfolding operators for circular domains with rapid oscillations with high amplitude of O(1) to study the homogenization of an elliptic problem.

The Kazdan-Warner equation on canonically compactifiable graphs

Abstract: We study the Kazdan–Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.

Spreading with two speeds and mass segregation in a diffusive competition system with free boundaries

Abstract: We investigate the spreading behavior of two invasive species modeled by a Lotka–Volterra diffusive competition system with two free boundaries in a spherically symmetric setting. We show that, for the weak–strong competition case, under suitable assumptions, both species in the system can successfully spread into the available environment, but their spreading speeds are different, and their population masses tend to segregate, with the slower spreading competitor having its population concentrating on an expanding ball, say $$B_t$$ , and the faster spreading competitor concentrating on a spherical shell outside $$B_t$$ that disappears to infinity as time goes to infinity.

$$L^p$$ L p Christoffel–Minkowski problem: the case $$1< p<k+1$$ 1 < p < k + 1

Abstract: We consider a fully nonlinear partial differential equation associated to the intermediate $$L^p$$ Christoffel–Minkowski problem in the case $$1

Nonlocal H-convergence

Abstract: We introduce the concept of nonlocal H-convergence. For this we employ the theory of abstract closed complexes of operators in Hilbert spaces. We show uniqueness of the nonlocal H-limit as well as a corresponding compactness result. Moreover, we provide a characterisation of the introduced concept, which implies that local and nonlocal H-convergence coincide for multiplication operators. We provide applications to both nonlocal and nonperiodic fully time-dependent 3D Maxwell’s equations on rough domains. The material law for Maxwell’s equations may also rapidly oscillate between eddy current type approximations and their hyperbolic non-approximated counter parts. Applications to models in nonlocal response theory used in quantum theory and the description of meta-materials, to fourth order elliptic problems as well as to homogenisation problems on Riemannian manifolds are provided.

Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equations

Abstract: We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison principle, which we refer to as the principle of propagation of touching points, for operators of the form $$ abla ^2 \psi + L(x,\psi , abla \psi )$$ which are non-decreasing in $$\psi $$ .

Planar least gradient problem: existence, regularity and anisotropic case

Abstract: We show existence of solutions to the least gradient problem on the plane for boundary data in $$BV(\partial \varOmega )$$ . We also provide an example of a function $$f \in L^1(\partial \varOmega ) \backslash $$ $$(C(\partial \varOmega ) \cup BV(\partial \varOmega ))$$ , for which the solution exists. We also show non-uniqueness of solutions even for smooth boundary data in the anisotropic case for a nonsmooth anisotropy. We additionally prove a regularity result valid also in higher dimensions.

General transport problems with branched minimizers as functionals of 1-currents with prescribed boundary

Abstract: A prominent model for transportation networks is branched transport, which seeks the optimal transportation scheme to move material from a given initial to a given final distribution. The cost of the scheme encodes a higher transport efficiency the more mass is moved together, which automatically leads to optimal transportation networks with a hierarchical branching structure. The two major existing model formulations use either mass fluxes (vector-valued measures, Eulerian formulation) or patterns (probabilities on the space of particle paths, Lagrangian formulation). In the branched transport problem the transportation cost is a fractional power of the transported mass. In this paper we instead analyse the much more general class of transport problems in which the transportation cost is merely a nonnegative increasing and subadditive function (in a certain sense this is the broadest possible generalization of branched transport). In particular, we address the problem of the equivalence of the above-mentioned formulations in this wider context. However, the newly-introduced class of transportation costs lacks strict concavity which complicates the analysis considerably. New ideas are required, in particular, it turns out convenient to state the problem via 1-currents. Our analysis also includes the well-posedness, some network properties, as well as a metrization and a length space property of the model cost, which were previously only known for branched transport. Some already existing arguments in that field are given a more concise and simpler form.

On the Hölder continuous subsolution problem for the complex Monge–Ampère equation

Abstract: We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge–Ampere equation admits Holder continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge–Ampere total mass, we obtain an affirmative answer to a question of Zeriahi et al. (Complex Var. Elliptic Equ. 61(7):902–930, 2016).

On the Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space

Abstract: Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.