Volume 15, Issue 4

In this paper, we prove borderline gradient continuity of viscosity solutions to fully nonlinear elliptic equations at the boundary of a C 1 , Dini C^{1,\mathrm{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 W 1 , q W^{1,q} estimates, our proof is slightly more geometric and is based on compactness arguments inspired by the techniques in the fundamental works of Luis Caffarelli.

For any compact Lie group 𝐺 and closed, smooth Riemannian manifold ( X , g ) (X,g) of dimension d ≥ 2 d\geq 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 L p L^{p} -small curvature, when p > d / 2 p>d/2 , to the case of a connection with L d / 2 L^{d/2} -small curvature. We prove an optimal Łojasiewicz–Simon gradient inequality for abstract Morse–Bott functions on Banach manifolds, generalizing an earlier result due to the author and Maridakis (2019), principally by removing the hypothesis that the Hessian operator be Fredholm with index zero. We apply this result to prove the optimal Łojasiewicz–Simon gradient inequality for the self-dual Yang–Mills energy function near regular anti-self-dual connections over closed Riemannian four-manifolds and for the full Yang–Mills energy function over closed Riemannian manifolds of dimension d ≥ 2 d\geq 2 , when known to be Morse–Bott at a given Yang–Mills connection. We also prove the optimal Łojasiewicz–Simon gradient inequality by direct analysis near a given flat connection that is a regular point of the curvature map.

We study the generalized eigenvalue problem in ℝ N {\mathbb{R}^{N}} 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 and compare them. We also discuss the maximum principles and uniqueness of principal eigenfunctions.

We investigate the Kazdan–Warner equation on a network. In this case, the differential equation is defined on each edge, while appropriate transition conditions of Kirchhoff type are prescribed at the vertices. We show that the whole Kazdan–Warner theory, both for the noncritical and the critical case, extends to the present setting.

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 ( GSBD p {\mathrm{GSBD}^{p}} ) 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, or material voids. Our approach is based on the global method for relaxation devised in [G. Bouchitté, I. Fonseca and L. Mascarenhas, A global method for relaxation, Arch. Ration. Mech. Anal. 145 1998, 1, 51–98] and a recent Korn-type inequality in GSBD p {\mathrm{GSBD}^{p}} , cf. [F. Cagnetti, A. Chambolle and L. Scardia, Korn and Poincaré–Korn inequalities for functions with a small jump set, preprint 2020]. Our general strategy also allows to generalize integral representation results in SBD p {\mathrm{SBD}^{p}} , obtained in dimension two [S. Conti, M. Focardi and F. Iurlano, Integral representation for functionals defined on SBD p \mathrm{SBD}^{p} in dimension two, Arch. Ration. Mech. Anal. 223 2017, 3, 1337–1374], to higher dimensions, and to revisit results in the framework of generalized special functions of bounded variation ( GSBV p {\mathrm{GSBV}^{p}} ).

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. In higher dimensions, we use the inverse mean curvature flow to obtain new Minkowski inequalities when the ambient radial Ricci curvature is constantly negative.

We consider the equation - Δ p ⁢ u = f ⁢ ( u ) {-\Delta_{p}u=f(u)} in a smooth bounded domain of ℝ n {\mathbb{R}^{n}} , where Δ p {\Delta_{p}} is the p -Laplace operator. Explicit examples of unbounded stable energy solutions are known if n ≥ p + 4 ⁢ p p - 1 {n\geq p+\frac{4p}{p-1}} . Instead, when n < p + 4 ⁢ p p - 1 {n<p+\frac{4p}{p-1}} , stable solutions have been proved to be bounded only in the radial case or under strong assumptions on f . In this article we solve a long-standing open problem: we prove an interior C α {C^{\alpha}} bound for stable solutions which holds for every nonnegative f ∈ C 1 {f\in C^{1}} whenever p ≥ 2 {p\geq 2} and the optimal condition n < p + 4 ⁢ p p - 1 {n<p+\frac{4p}{p-1}} holds. When p ∈ ( 1 , 2 ) {p\in(1,2)} , we obtain the same result under the nonsharp assumption n < 5 ⁢ p {n<5p} . These interior estimates lead to the boundedness of stable and extremal solutions to the associated Dirichlet problem when the domain is strictly convex. Our work extends to the p -Laplacian some of the recent results of Figalli, Ros-Oton, Serra, and the first author for the classical Laplacian, which have established the regularity of stable solutions when p = 2 {p=2} in the optimal range n < 10 {n<10} .

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. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], all the main properties and features (e.g. coarea, Leibniz, and Gauss–Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in BV \mathrm{BV} . We remark that the standard pairing in general does not share this property.

We investigate the different notions of solutions to the double-phase equation - div ⁡ ( | D ⁢ u | p - 2 ⁢ D ⁢ u + a ⁢ ( x ) ⁢ | D ⁢ u | q - 2 ⁢ D ⁢ u ) = 0 , -{\operatorname{div}(\lvert Du\rvert^{p-2}Du+a(x)\lvert Du\rvert^{q-2}Du)}=0, which is characterized by the fact that both ellipticity and growth switch between two different types of polynomial according to the position. We introduce the 𝒜 H ⁢ ( ⋅ ) {\mathcal{A}_{H(\,{\cdot}\,)}} -harmonic functions of nonlinear potential theory and then show that 𝒜 H ⁢ ( ⋅ ) {\mathcal{A}_{H(\,{\cdot}\,)}} -harmonic functions coincide with the distributional and viscosity solutions, respectively. This implies that the distributional and viscosity solutions are exactly the same.

There still exist many unsolved problems on the study related to John–Nirenberg spaces. In this article, the authors introduce two new vanishing subspaces of the John–Nirenberg space JN p ⁢ ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} denoted, respectively, by VJN p ⁢ ( ℝ n ) {\mathrm{VJN}_{p}(\mathbb{R}^{n})} and CJN p ⁢ ( ℝ n ) {\mathrm{CJN}_{p}(\mathbb{R}^{n})} , and establish their equivalent characterizations which are counterparts of those characterizations for the classic spaces VMO ⁢ ( ℝ n ) {\mathrm{VMO}(\mathbb{R}^{n})} and CMO ⁢ ( ℝ n ) {\mathrm{CMO}(\mathbb{R}^{n})} obtained, respectively, by D. Sarason and A. Uchiyama. All these results shed some light on the mysterious space JN p ⁢ ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} . The approach strongly depends on the fine geometrical properties of dyadic cubes, which enable the authors to subtly classify any collection of interior pairwise disjoint cubes.

We consider the PDE - Δ p ⁢ u = ρ -\Delta_{p}u=\rho , where 𝜌 is a signed Borel measure on R n \mathbb{R}^{n} . For each p > n p>n , we characterize solutions as extremals of a generalized Morrey inequality determined by 𝜌.

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 we generalize certain rigidity properties of divergence-free vector fields to vector-valued measures. Namely, we show that if a locally finite vector-valued measure has zero divergence, vanishes in the lower half-space and the normal component of the unit tangent vector of the measure is bounded from below (in the upper half-plane), then the measure is identically zero.

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 of quasiminimizing sequences, preprint 2020, https://arxiv.org/abs/2002.08876] which allows to take into account the free boundary. Moreover, we show that the Reifenberg class is closed under weak convergence without restriction on the coefficient group.

In this paper, we characterize the equilibrium measure for a family of nonlocal and anisotropic energies I α I_{\alpha} that describe the interaction of particles confined in an elliptic subset of the plane. The case α = 0 \alpha=0 corresponds to purely Coulomb interactions, while the case α = 1 \alpha=1 describes interactions of positive edge dislocations in the plane. The anisotropy into the energy is tuned by the parameter 𝛼 and favors the alignment of particles. We show that the equilibrium measure is completely unaffected by the anisotropy and always coincides with the optimal distribution in the case α = 0 \alpha=0 of purely Coulomb interactions, which is given by an explicit measure supported on the boundary of the elliptic confining domain. Our result does not seem to agree with the mechanical conjecture that positive edge dislocations at equilibrium tend to arrange themselves along “wall-like” structures. Moreover, this is one of the very few examples of explicit characterization of the equilibrium measure for nonlocal interaction energies outside the radially symmetric case.