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

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

Journal Name: Advances in Calculus of Variations Volume: 14 Issue: 1 Pages: i-iv

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.

For any fixed integer D>1 we show that there exists M∈[e-1,1] such that for any open, bounded, convex domain Ω⊂ℝD 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,∞). Moreover, for any real number s>M there exists an open, bounded, convex domain Ω⊂ℝD

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

We consider different fractional Neumann Laplacians of order s∈(0,1) on domains Ω⊂ℝn, namely, the restricted Neumann Laplacian(-ΔΩN)Rs, the semirestricted Neumann Laplacian(-ΔΩN)Srs and the spectral Neumann Laplacian(-ΔΩN)Sps. In particular, we are interested in the attainability of Sobolev constants for these operators when Ω is a half-space.

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 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

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

We characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the L2-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded

We study the branch set of a mapping between subsets of ℝn, i.e., the set where a given mapping is not defining a local homeomorphism. We construct several sharp examples showing that the branch set or its image can have positive measure.

The present paper arose from recent studies of energy-minimal deformations of planar domains. We are concerned with the Dirichlet energy. In general the minimal mappings need not be homeomorphisms. In fact, a part of the domain near its boundary may collapse into the boundary of the target domain. In mathematical models of nonlinear elasticity this is interpreted as interpenetration of matter. We call

Integrals of the Calculus of Variations with p,q-growth may have not smooth minimizers, not even bounded, for general p,q exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f⁢(x,ξ) with dependence on the modulus of the gradient, i.e. f⁢(x,ξ)=g⁢(x,|ξ|). Without imposing structure conditions, we prove

Let H be a norm of ℝN and H0 the dual norm of H. Denote by ΔH the Finsler–Laplace operator defined by ΔH⁢u:=div⁡(H⁢(∇⁡u)⁢∇ξ⁡H⁢(∇⁡u)). In this paper we prove that the Finsler–Laplace operator ΔH acts as a linear operator to H0-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation

We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding

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) of a Hamiltonian H, with ε>0. We establish the convergence, as ε→0+, of solutions of the Hamilton–Jacobi equations and identify the limit of the solutions as the solution of systems of ordinary

In the present paper we consider the Carnot–Carathéodory distance δE to a closed set E in the sub-Riemannian Heisenberg groups ℍn, n⩾1. The ℍ-regularity of δE is proved under mild conditions involving a general notion of singular points. In case E is a Euclidean Ck submanifold, k⩾2, we prove that δE is Ck out of the singular set. Explicit expressions for the volume of the tubular neighborhood when

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