# Calculus of Variations and Partial Differential Equations

## List of Papers (Total 89)

#### Regular 1-harmonic flow

We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e., the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target...

#### Metric currents and the Poincaré inequality

We show that a complete doubling metric space $$(X,d,\mu )$$ supports a weak 1-Poincaré inequality if and only if it admits a pencil of curves (PC) joining any pair of points $$s,t \in X$$. This notion was introduced by S. Semmes in the 90’s, and has been previously known to be a sufficient condition for the weak 1-Poincaré inequality. Our argument passes through the intermediate...

#### Partial regularity for manifold constrained p(x)-harmonic maps

We prove that manifold constrained p(x)-harmonic maps are locally $$C^{1,\beta _{0}}$$-regular outside a set of zero n-dimensional Lebesgue’s measure, for some $$\beta _{0} \in (0,1)$$. We also provide an estimate from above of the Hausdorff dimension of the singular set.

#### Positive solutions for nonlinear singular elliptic equations of p-Laplacian type with dependence on the gradient

In this paper, we study a nonlinear Dirichlet problem of p-Laplacian type with combined effects of nonlinear singular and convection terms. An existence theorem for positive solutions is established as well as the compactness of solution set. Our approach is based on Leray–Schauder alternative principle, method of sub-supersolution, nonlinear regularity, truncation techniques...

#### On the geometry of geodesics in discrete optimal transport

We consider the space of probability measures on a discrete set $$\mathcal {X}$$, endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset $$\mathcal {Y}\subseteq \mathcal {X}$$, it is natural to ask whether they can be connected by a constant speed geodesic with support in $$\mathcal {Y}$$ at all times. Our main result answers this...

#### Spectral estimates for infinite quantum graphs

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections...

#### Bifurcation from infinity for elliptic problems on $${\mathbb {R}}^N$$

In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schrödinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the parameter is an eigenvalue of the hamiltonian lying below the asymptotic bottom of the bounded part of the potential. Thus the bifurcating solution...

#### Conservation of energy for the Euler–Korteweg equations

In this article we study the principle of energy conservation for the Euler–Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler–Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.

#### Ground states in the diffusion-dominated regime

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...

#### Analysis of boundary bubbles for almost minimal cylinders

We analyse the asymptotic behaviour of solutions of the Teichmüller harmonic map flow from cylinders, and more generally of ‘almost minimal cylinders’, in situations where the maps satisfy a Plateau-boundary condition for which the three-point condition degenerates. We prove that such a degenerating boundary condition forces the domain to stretch out as a boundary bubble forms...

#### Global existence of Dirac-wave maps with curvature term on expanding spacetimes

We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions.

#### Ergodic mean field games with Hörmander diffusions

We prove existence of solutions for a class of systems of subelliptic PDEs arising from mean field game systems with Hörmander diffusion. These results are motivated by the feedback synthesis mean field game solutions and the nash equilibria of a large class of N-player differential games.

#### Planar least gradient problem: existence, regularity and anisotropic case

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...

#### Existence of solutions to a general geometric elliptic variational problem

We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed m dimensional subsets of $${\mathbf {R}}^n$$ which is stable under taking smooth deformations homotopic to the identity and under local Hausdorff limits. We prove that the minimiser exists inside the class and is an $$({\mathscr {H}}^m,m)$$ rectifiable set in the sense of...

#### Partial regularity for a nonlinear sigma model with gravitino in higher dimensions

We study the regularity problem of the nonlinear sigma model with gravitino fields in higher dimensions. After setting up the geometric model, we derive the Euler–Lagrange equations and consider the regularity of weak solutions defined in suitable Sobolev spaces. We show that any weak solution is actually smooth under some smallness assumption for certain Morrey norms. By...

#### Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast...

#### Geometric inequalities on Heisenberg groups

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 Schmuckenschläger. The...

#### Bifurcations from the orbit of solutions of the Neumann problem

The purpose of this paper is to study weak solutions of a nonlinear Neumann problem considered on a ball. Assuming that the potential is invariant, we consider an orbit of critical points, i.e. we do not assume that critical points are isolated. We apply techniques of equivariant analysis to examine bifurcations from the orbits of trivial solutions. We formulate sufficient...

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

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

#### Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps $$(u_n,v_n)$$ (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the...

#### Stable self-similar blowup in the supercritical heat flow of harmonic maps

We consider the heat flow of corotational harmonic maps from $$\mathbb {R}^3$$ to the three-sphere and prove the nonlinear asymptotic stability of a particular self-similar shrinker that is not known in closed form. Our method provides a novel, systematic, robust, and constructive approach to the stability analysis of self-similar blowup in parabolic evolution equations. In...

#### Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain $$D \subset \mathbb {R}^d$$ with $$d \ge 2$$, for every (measurable) uniformly elliptic tensor field a and for almost every point $$y \in D$$, there exists a unique Green’s function centred in y associated to the vectorial operator $$-\nabla \cdot a\nabla$$ in D. This result...

#### Minkowski valuations on convex functions

A classification of $${\text {SL}}(n)$$ contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new $${\text {SL}}(n)$$ covariant Minkowski valuation on convex functions is defined and characterized.

#### Coarse regularity of solutions to a nonlinear sigma-model with  $$L^p$$  gravitino

The regularity of weak solutions of a two-dimensional nonlinear sigma model with coarse gravitino is shown. Here the gravitino is only assumed to be in $$L^p$$ for some $$p>4$$. The precise regularity results depend on the value of p.