Article Frontmatter was published on April 1, 2021 in the journal Advances in Calculus of Variations (volume 14, issue 2).

Using the optimal mass transport method and a suitable quasi-conformal mapping, we study the sharp weighted isoperimetric, Sobolev, Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg inequalities. The class of weight functions under consideration includes all nonnegative homogeneous weights satisfying a concavity condition that is equivalent to a usual curvature-dimension bound and the nonnegativity

In the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of BV{\mathrm{BV}} functions with zero boundary values. In particular, we show that the class is the closure of compactly supported BV{\mathrm{BV}} functions in the BV{\mathrm{BV}} norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz

We consider a weakly coupled system of discounted Hamilton–Jacobi equations set on a closed Riemannian manifold. We prove that the corresponding solutions converge to a specific solution of the limit system as the discount factor goes to 0. The analysis is based on a generalization of the theory of Mather minimizing measures for Hamilton–Jacobi systems and on suitable random representation formulae

We define a class of Sobolev W1,p⁢(Ω,ℝn){W^{1,p}(\Omega,\mathbb{R}^{n})} functions, with p>n-1{p>n-1}, such that its trace on ∂⁡Ω{\partial\Omega} is also Sobolev, and do not present cavitation in the interior or on the boundary. We show that if a function in this class has positive Jacobian and coincides on the boundary with an injective map, then the function is itself injective. We then prove the

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda Jr. We show that these two notions coincide if the measure is doubling and supports a 1-Poincaré inequality. In doing so, we also prove that if the measure is doubling and supports a 1-Poincaré inequality, then the metric space supports a Semmes family of curves

In this article we study two classical problems in convex geometry associated to 𝒜{\mathcal{A}}-harmonic PDEs, quasi-linear elliptic PDEs whose structure is modelled on the p -Laplace equation. Let p be fixed with 2≤n≤p<∞{2\leq n\leq p<\infty}. For a convex compact set E in ℝn{\mathbb{R}^{n}}, we define and then prove the existence and uniqueness of the so-called 𝒜{\mathcal{A}}-harmonic Green’s function

For any compact Lie group 𝐺 and closed, smooth Riemannian manifold (X,g) of dimension d≥2, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal 𝐺-bundle over 𝑋 supporting a connection with Lp-small curvature, when p>d/2, to the case of a connection with Ld/2-small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott

In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies Iα that describe the interaction of particles confined in an elliptic subset of the plane. The case α=0 corresponds to purely Coulomb interactions, while the case α=1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter 𝛼 and

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton–Jacobi equations depending nonlinearly on the unknown function. Let H⁢(x,p,u) be a continuous Hamiltonian which is strictly increasing in u, and is convex and coercive in p. For each parameter λ>0, we denote by uλ the unique viscosity solution of the Hamilton–Jacobi equation

We consider the structure of divergence-free vector measures on the plane. We show that such measures can be decomposed into measures induced by closed simple curves. More generally, we show that if the divergence of a planar vector-valued measure is a signed measure, then the vector-valued measure can be decomposed into measures induced by simple curves (not necessarily closed). As an application

For any fixed integer D>1{D>1} we show that there exists M∈[e-1,1]{M\in[e^{-1},1]} such that for any open, bounded, convex domain Ω⊂ℝD{\Omega\subset{\mathbb{R}}^{D}} with smooth boundary for which the maximum of the distance function to the boundary of Ω is less than or equal to M , the principal frequency of the p -Laplacian on Ω is an increasing function of p on (1,∞){(1,\infty)}. Moreover, for any

We consider different fractional Neumann Laplacians of order s∈(0,1){s\in(0,1)} on domains Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}, namely, the restricted Neumann Laplacian (-ΔΩN)Rs{{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{R}}}}, the semirestricted Neumann Laplacian (-ΔΩN)Srs{{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sr}}}} and the spectral Neumann Laplacian (-ΔΩN)Sps{{(-\Delta_{\Omega}^{N})^{s}_{\mathrm{Sp}}}}.

The notion of weakly monotone functions extends the classical definition of monotone function, that can be traced back to Lebesgue. It was introduced, in the framework of Sobolev spaces, by Manfredi, in connection with the analysis of the regularity of maps of finite distortion appearing in the theory of nonlinear elasticity. Diverse authors, including Iwaniecz, Kauhanen, Koskela, Maly, Onninen, Zhong

In this work we prove convergence results of sequences of Riemannian 4-manifolds with almost vanishing L2{L^{2}}-norm of a curvature tensor and a non-collapsing bound on the volume of small balls. In Theorem 1.1 we consider a sequence of closed Riemannian 4-manifolds, whose L2{L^{2}}-norm of the Riemannian curvature tensor tends to zero. Under the assumption of a uniform non-collapsing bound and a

We investigate the asymptotic behavior of solutions of Hamilton–Jacobi equations with large Hamiltonian drift terms in an open subset of the two-dimensional Euclidean space. The drift is given by ε-1⁢(Hx2,-Hx1){{\varepsilon}^{-1}(H_{x_{2}},-H_{x_{1}})} of a Hamiltonian H , with ε>0{{\varepsilon}>0}. We establish the convergence, as ε→0+{{\varepsilon}\to 0+}, of solutions of the Hamilton–Jacobi equations

We consider second-order divergence form elliptic operators with W1,1{W^{1,1}} coefficients, in a uniform domain Ω with Ahlfors regular boundary. We show that the A∞{A_{\infty}} property of the elliptic measure associated to any such operator implies that Ω is a set of locally finite perimeter whose boundary, ∂⁡Ω{\partial\Omega}, is rectifiable. As a corollary we show that for this type of operators

In the present paper we consider the Carnot–Carathéodory distance δE{\delta_{E}} to a closed set E in the sub-Riemannian Heisenberg groups ℍn{{\mathbb{H}}^{n}}, n⩾1{n\geqslant 1}. The ℍ{{\mathbb{H}}}-regularity of δE{\delta_{E}} is proved under mild conditions involving a general notion of singular points. In case E is a Euclidean Ck{C^{k}} submanifold, k⩾2{k\geqslant 2}, we prove that δE{\delta_{E}}

Article Frontmatter was published on January 1, 2021 in the journal Advances in Calculus of Variations (volume 14, issue 1).

We investigate the different notions of solutions to the double-phase equation

In this paper we study the (BV,Lp)-decomposition, p=1,2, of functions in metric random walk spaces, a general workspace that includes weighted graphs and nonlocal models used in image processing. We obtain the Euler-Lagrange equations of the corresponding variational problems and their gradient flows. In the case p=1 we also study the associated geometric problem and the thresholding parameters describing

We study the generalized eigenvalue problem in ℝN for a general convex nonlinear elliptic operator which is locally elliptic and positively 1-homogeneous. Generalizing [H. Berestycki and L. Rossi, Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains, Comm. Pure Appl. Math. 68 2015, 6, 1014–1065], we consider three different notions of generalized eigenvalues

We solve two variants of the Reifenberg problem for all coefficient groups. We carry out the direct method of the calculus of variation and search a solution as a weak limit of a minimizing sequence. This strategy has been introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi and allowed them to solve the Reifenberg problem. We use an analogous strategy proved in [C. Labourie, Weak limits

In this paper we prove an integral representation formula for a general class of energies defined on the space of generalized special functions of bounded deformation (GSBDp) in arbitrary space dimensions. Functionals of this type naturally arise in the modeling of linear elastic solids with surface discontinuities including phenomena as fracture, damage, surface tension between different elastic phases

Building upon the recent results in [M. Focardi and E. Spadaro, On the measure and the structure of the free boundary of the lower-dimensional obstacle problem, Arch. Ration. Mech. Anal. 230 2018, 1, 125–184] we provide a thorough description of the free boundary for the solutions to the fractional obstacle problem in ℝn+1 with obstacle function φ (suitably smooth and decaying fast at infinity) up

This paper deals with locally constrained inverse curvature flows in a broad class of Riemannian warped spaces. For a certain class of such flows, we prove long-time existence and smooth convergence to a radial coordinate slice. In the case of two-dimensional surfaces and a suitable speed, these flows enjoy two monotone quantities. In such cases, new Minkowski type inequalities are the consequence

We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler–Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by

We consider the equation -Δp⁢u=f⁢(u) in a smooth bounded domain of ℝn, where Δp is the p-Laplace operator. Explicit examples of unbounded stable energy solutions are known if n≥p+4⁢pp-1. Instead, when n

Journal Name: Advances in Calculus of Variations Volume: 13 Issue: 4 Pages: i-iv

In this paper, we prove borderline gradient continuity of viscosity solutions to fully nonlinear elliptic equations at the boundary of a C1,Dini-domain. Our main result constitutes the boundary analogue of the borderline interior gradient regularity estimates established by P. Daskalopoulos, T. Kuusi and G. Mingione. We however mention that, differently from the approach used there which is based on

In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function ∥y-x∥sc in Rn or 𝑇 is an optimal transport map for the Monge problem with cost function d⁢(x,y), the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity

We introduce a family of pairings between a bounded divergence-measure vector field and a function u of bounded variation, depending on the choice of the pointwise representative of u. We prove that these pairings inherit from the standard one, introduced in [G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q

We prove a homogenization theorem for quadratic convolution energies defined in perforated domains. The corresponding limit is a Dirichlet-type quadratic energy, whose integrand is defined by a non-local cell-problem formula. The proof relies on an extension theorem from perforated domains belonging to a wide class containing compact periodic perforations.

We consider the constant Q-curvature metric problem in a given conformal class on a conic 4-manifold and study related differential equations. We define subcritical, critical, and supercritical conic 4-manifolds. Following [M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 1991, 2, 793–821] and [S.-Y. A. Chang and P. C. Yang, Extremal metrics

Journal Name: Advances in Calculus of Variations Volume: 13 Issue: 3 Pages: i-iv

The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model n-dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016)

We study the comparison principle for non-negative solutions of the equation

We introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss

We show how to derive (variants of) Michell truss theory in two and three dimensions rigorously as the vanishing weight limit of optimal design problems in linear elasticity in the sense of Γ-convergence. We improve our results from [H. Olbermann, Michell trusses in two dimensions as a Γ-limit of optimal design problems in linear elasticity, Calc. Var. Partial Differential Equations 56 2017, 6, Article

In this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as p→∞. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for

In this paper, we prove time estimates for solutions to a general nonhomogeneous parabolic problem whose operator satisfies nonstandard growth conditions. We also study the asymptotic behaviour of solutions to an anisotropic problem.

Chen, Torres and Ziemer ([], 2009) proved the validity of generalized Gauss–Green formulas and obtained the existence of interior and exterior normal traces for essentially bounded divergence measure fields on sets of finite perimeter using an approximation theory through sets with a smooth boundary. However, it is known that the proof of a crucial approximation lemma contained a gap. Taking inspiration

The dominative p-Laplace operator is introduced. This operator is a relative to the p-Laplacian, but with the distinguishing property of being sublinear. It explains the superposition principle in the p-Laplace equation.

We study the Wasserstein distance between two measures μ,ν which are mutually singular. In particular, we are interested in minimization problems of the form

In this paper we initiate the study of second-order variational problems in L∞, seeking to minimise the L∞ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given H∈C1⁢(ℝsn×n), for the functional

Journal Name: Advances in Calculus of Variations Volume: 13 Issue: 2 Pages: i-iv

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only

In recent years, many papers have been devoted to the regularity of doubly nonlinear singular evolution equations. Many of the proofs are unnecessarily complicated, rely on superfluous assumptions or follow an inappropriate approximation procedure. This makes the theory unclear and quite chaotic to a nonspecialist. The aim of this paper is to fix all the misprints, to follow correct procedures, to

We provide a unified strategy to show that solutions of dynamic programming principles associated to the p-Laplacian converge to the solution of the corresponding Dirichlet problem. Our approach includes all previously known cases for continuous and discrete dynamic programming principles, provides new results, and gives a convergence proof free of probability arguments.

We introduce and study certain variants of Gamow’s liquid drop model in which an anisotropic surface energy replaces the perimeter. After existence and nonexistence results are established, the shape of minimizers is analyzed. Under suitable regularity and ellipticity assumptions on the surface tension, Wulff shapes are minimizers in this problem if and only if the surface energy is isotropic. In sharp

We consider functionals of the form

In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert–Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in ℝn. Following the analysis for the planar case presented in [M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar

For curves of prescribed length embedded into the unit disk in two dimensions, we obtain scaling results for the minimal elastic energy as the length just exceeds 2⁢π and in the large length limit. In the small excess length case, we prove convergence to a fourth-order obstacle-type problem with integral constraint on the real line which we then solve. From the solution, we obtain the energy expansion

We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ⁢(t) is a non-negative convex function vanishing only at t=0. We show that this property is always satisfied in dimension n=2, while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ⁢(t)=c⁢t2) in dimension n≥4

In this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set Ω⊂ℝn and given p>1, we study the blow-up of functions u∈GSBV⁢(Ω), whose jump sets belong to an appropriate class 𝒥p and whose approximate gradients are p-th power summable. In analogy with the theory of p-capacity in the context of Sobolev spaces, we prove

In this paper, we consider a large class of Bernoulli-type free boundary problems with mixed periodic-Dirichlet boundary conditions. We show that solutions with non-flat profile can be found variationally as global minimizers of the classical Alt–Caffarelli energy functional.

In this work we study regularity properties of solutions to fractional elliptic problems with mixed Dirichlet–Neumann boundary data when dealing with the spectral fractional Laplacian.

We prove the existence of weak solutions to the homogeneous wave equation on a suitable class of time-dependent domains. Using the approach suggested by De Giorgi and developed by Serra and Tilli, such solutions are approximated by minimizers of suitable functionals in space-time.

The purpose of this paper is to study the existence of weak solutions for some classes of Schrödinger equations defined on the Euclidean space ℝd (d≥3). These equations have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using the Palais principle of symmetric criticality and a group-theoretical approach used on a suitable closed subgroup of

This paper deals with the existence of multiple solutions for the quasilinear equation