We introduce the concept of \(C^{m,\alpha }\)-nonlocal operators, extending the notion of second order elliptic operator in divergence form with \(C^{m,\alpha }\)-coefficients. We then derive the nonlocal analogue of the key existing results for elliptic equations in divergence form, notably the Hölder continuity of the gradient of the solutions in the case of \(C^{0,\alpha }\)-coefficients and the

Zhikov showed 1986 with his famous checkerboard example that functionals with variable exponents can have a Lavrentiev gap. For this example it was crucial that the exponent had a saddle point whose value was exactly the dimension. In 1997 he extended this example to the setting of the double phase potential. Again it was important that the exponents crosses the dimensional threshold. Therefore, it

Consider the Lamé operator \({\mathcal {L}}({\mathbf {u}}) :=\mu \Delta {\mathbf {u}}+(\lambda +\mu ) \nabla (\nabla \cdot {\mathbf {u}} )\) that arises in the theory of linear elasticity. This paper studies the geometric properties of the (generalized) Lamé eigenfunction \({\mathbf {u}}\), namely \(-{\mathcal {L}}({\mathbf {u}})=\kappa {\mathbf {u}}\) with \(\kappa \in {\mathbb {R}}_+\) and \({\mathbf

Throughout this note, we use the notation from

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak–Wenger

In this paper we study implicit obstacle problems driven by a nonhomogenous differential operator, called double phase operator, and a multivalued term which is described by Clarke’s generalized gradient. Based on a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis, we prove the existence of at least one solution.

In this paper the dependence of the best constants in Sobolev and Gagliardo–Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of \(\mathbb {R}^n\), Heisenberg, and general stratified Lie groups, in all these cases the results being new. The Sobolev norms may be defined in terms of

In any dimension \(N\ge 1\) and for given mass \(m>0\), we revisit the nonlinear scalar field equation with an \(L^2\) constraint: $$\begin{aligned} \left\{ \begin{aligned} -\Delta u&=f(u)-\mu u\quad \text {in}~\mathbb {R}^N,\\ \Vert u\Vert ^2_{L^2(\mathbb {R}^N)}&=m,\\ u&\in H^1(\mathbb {R}^N), \end{aligned} \right. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad

In this paper, we consider the contracting curvature flows of smooth closed surfaces in 3-dimensional hyperbolic space and in 3-dimensional sphere. In the hyperbolic case, we show that if the initial surface \(M_0\) has positive scalar curvature, then along the flow by a positive power \(\alpha \) of the mean curvature H, the evolving surface \(M_t\) has positive scalar curvature for \(t>0\). By assuming

We provide quantitative gradient bounds for solutions to certain parabolic equations with unbalanced polynomial growth and non-smooth coefficients.

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann–Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency \(\omega \in (-\infty ,0)\) is characterized as a global minimizer of the quadratic part of energy constrained

This paper determines when the Cauchy problem $$\begin{aligned} \left\{ \begin{array}{ll} {{\partial _t u} = \Delta u -Vu+ Wu^p} &{}\quad \text{ in } M \times (0, \infty ) \\ {u(\cdot ,0)= {u_0(\cdot )}} &{}\quad \text{ in } M \end{array} \right. \end{aligned}$$ has no global positive solution on a connected non-compact geodesically complete Riemannian manifold for a given triple (V, W, p). As the

We consider a nonlinear degenerate parabolic equation of porous medium type, whose diffusion is driven by the (spectral) fractional Laplacian on the hyperbolic space. We provide existence results for solutions, in an appropriate weak sense, for data belonging either to the usual \(L^p\) spaces or to larger (weighted) spaces determined either in terms of a ground state of \(\Delta _{\mathbb {H}^{N}}\)

We consider the following problem: minimize the functional \(\int _\Omega f(\nabla u(x))\, dx\) in the class of concave functions \(u: \Omega \rightarrow [0,M]\), where \(\Omega \subset {\mathbb {R}}^2\) is a convex body and \(M > 0\). If \(f(x) = 1/(1 + |x|^2)\) and \(\Omega \) is a circle, the problem is called Newton’s problem of least resistance. It is known that the problem admits at least one

We study singular radially symmetric solution to the Lin–Ni–Takagi equation for a supercritical power non-linearity in dimension \(N\ge 3\). It is shown that for any ball and any \(k \ge 0\), there is a singular solution that satisfies Neumann boundary condition and oscillates at least k times around the constant equilibrium. Moreover, we show that the Morse index of the singular solution is finite

We define a hybrid between Ollivier and Bakry-Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue estimates follow from the standard arguments. We prove gradient estimates similar to the ones obtained from Bakry-Emery curvature allowing us to prove

We study the Ricci flow on \({\mathbb {R}}^{4}\) starting at an SU(2)-cohomogeneity 1 metric \(g_{0}\) whose restriction to any hypersphere is a Berger metric. We prove that if \(g_{0}\) has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension \(n

In quantitative genetics, viscosity solutions of Hamilton–Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function I(t) that arises as the counterpart of a non-negativity constraint on the solution at each time. Although the uniqueness of viscosity solutions is known for many

Let \(G=(V,E)\) be a finite connected graph, and let \(\kappa : V\rightarrow {\mathbb {R}}\) be a function such that \(\int _V\kappa \mathrm{d}\mu <0\). We consider the following Kazdan–Warner equation on G: $$\begin{aligned} \varDelta u+\kappa -K_\lambda e^{2u}=0, \end{aligned}$$ where \(K_\lambda =K+\lambda \) and \(K: V\rightarrow {\mathbb {R}}\) is a non-constant function satisfying \(\max _{x\in

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension 3. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary \(C^2\) surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable

We establish sharp \(C_{\text {loc}}^{1, \beta }\) geometric regularity estimates for bounded solutions of a class of fully nonlinear elliptic equations with non-homogeneous degeneracy, whose model equation is given by $$\begin{aligned} \left[ |Du|^p+\mathfrak {a}(x)|Du|^q\right] {\mathcal {M}}_{\lambda , \Lambda }^{+}(D^2 u)= f(x, u) \quad \text {in} \quad \Omega , \end{aligned}$$ for a bounded and

This paper is concerned with the existence and uniqueness of transition fronts of a general reaction-diffusion-advection equation in domains with multiple branches. In this paper, every branch in the domain is not necessary to be straight and we use the notions of almost-planar fronts to generalize the standard planar fronts. Under some assumptions of existence and uniqueness of almost-planar fronts

We study the limiting behavior as \(|x|\rightarrow \infty \) of extremal functions u for Morrey’s inequality on \(\mathbb {R}^n\). In particular, we compute the limit of u(x) as \(|x|\rightarrow \infty \) and show |x||Du(x)| tends to 0. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form \(-\Delta _pu=c(\delta _{x_0}-\delta _{y_0})\) for

We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton–Jacobi–Bellman equation and a forward in time Fokker–Planck equation. The value function u is continuous and satisfies general Kirchhoff conditions at the vertices. The density m of the distribution of states satisfies dual transmission conditions:

In this paper we study the boundary value problem for the equation \(\text{ div }\left( D(\nabla u)\nabla \left( \text{ div }\left( |\nabla u|^{p-2}\nabla u+\beta \frac{\nabla u}{|\nabla u|}\right) \right) \right) +au=f\) in the \(z=(x,y)\) plane. This problem is derived from a continuum model for the relaxation of a crystal surface below the roughing temperature. The mathematical challenge is of twofolds

We address the questions (P1), (P2) asked in Kirchheim et al. (Studying nonlinear PDE by geometry in matrix space. Geometric analysis and nonlinear partial differential equations, Springer, Berlin, 1986) concerning the structure of the Rank-1 convex hull of a submanifold \(\mathcal {K}_1\subset M^{3\times 2}\) that is related to weak solutions of the two by two system of Lagrangian equations of elasticity

We apply topological methods and a Lusternik-Schnirelmann-type approach to prove existence results for closed geodesics of Finsler metrics on spheres and projective spaces. The main tool in the proofs are spherical complexities, which have been introduced in earlier work of the author. Using them, we show how pinching conditions and inequalities between a Finsler metric and a globally symmetric metric

The Stokes resolvent problem \(\lambda u - \Delta u + \nabla \phi = f\) with \({\text {div}}(u) = 0\) subject to homogeneous Dirichlet or homogeneous Neumann-type boundary conditions is investigated. In the first part of the paper we show that for Neumann-type boundary conditions the operator norm of \(\mathrm {L}^2_{\sigma } (\Omega ) \ni f \mapsto \phi \in \mathrm {L}^2 (\Omega )\) decays like \(|\lambda

Consider the diffusive Hamilton–Jacobi equation $$\begin{aligned} u_t-\Delta u=|\nabla u|^p+h(x)\ \ \text { in } \Omega \times (0,T) \end{aligned}$$ with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the question of the gradient blowup rate for classical solutions with \(p>2\). We first consider the case of time-increasing solutions. For such

We consider a class of “wild” initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural \(L^1\)-topology and show that its complement is rather large, specifically it is an open dense set.

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) k-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for \(k\ge 3\). For \(k=1\) the classical result of Weinstock (J Ration Mech Anal 3:745–753, 1954) shows that \(\sigma _1\)

We investigate the geometric properties of Steklov eigenfunctions in smooth manifolds. We derive the refined doubling estimates and Bernstein’s inequalities. For the real analytic manifolds, we are able to obtain the sharp upper bound for the measure of interior nodal sets.

In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional E. We show that it has the longtime existence and subconverges to spherical caps. As an application, we solve an isoperimetric problem for hypersurfaces with capillary boundary

The purpose of the present paper is to study a class of semilinear elliptic Dirichlet boundary value problems in the ball, where the nonlinearities involve the sum of a sublinear variable exponent and a superlinear (may be supercritical) variable exponents of the form \(0\le f(r,u)\le a_1|u|^{p(r)-1}\), if \(u\ge 0\), where \(r = |x|\), \(p(r ) = 2^* + r^{\alpha }\), with \(\alpha > 0\), and \(2^*

We study the equation \(-\text {div}(A(x,\nabla u))=|u|^{q_1-1}u|\nabla u|^{q_2}+\mu \) where \(A(x,\nabla u)\sim |\nabla u|^{p-2}\nabla u\) in some suitable sense, \(\mu \) is a measure and \(q_1\), \(q_2\) are nonnegative real numbers and satisfy \(q_1+q_2>p-1\). We give sufficient conditions for existence of solutions expressed in terms of the Wolff potential or the Riesz potentials of the measure

For the coupled Schrödinger system $$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+V_1(x)u=\mu _1u^3+\beta uv^2\ \ \text{ in }\ {{\mathbb {R}}}^4,\\&-\Delta v+V_2(x)v=\beta u^2v+\mu _2v^3\ \ \,\text{ in }\ {{\mathbb {R}}}^4,\\&\ u\ge 0,\ v\ge 0\ \ \text{ in }\ {{\mathbb {R}}}^4,\ \ u, v\in D^{1,2}({{\mathbb {R}}}^4), \end{aligned}\right. \end{aligned}$$ where \(V_1, V_2\in L^2({{\mathbb {R}}}^4)\

We determine bubble tree convergence for a sequence of harmonic maps, with uniform energy bounds, from a compact Riemann surface into a compact locally CAT(1) space. In particular, we demonstrate energy quantization and the no-neck property for such a sequence. In the smooth setting, Jost (Two-dimensional geometric variational problems. Pure and applied mathematics. Wiley, New York, 1991) and Parker

We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition.

For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. We introduce a new generalization for strongly connected directed graphs by using the mean transition probability kernel which appears in the formulation

We study the normalized solutions of the fractional nonlinear Schrödinger equations with combined nonlinearities $$\begin{aligned} (-\Delta )^s u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u \quad \text{ in }~{\mathbb {R}}^N, \end{aligned}$$ and we look for solutions which satisfy prescribed mass $$\begin{aligned} \int _{\mathbb {R}^N}|u|^2=a^2, \end{aligned}$$ where \(N\ge 2,s\in (0,1),\mu \in \mathbb {R}\)

We study the vanishing discount problem for a nonlinear monotone system of Hamilton–Jacobi equations. This continues the first author’s investigation on the vanishing discount problem for a monotone system of Hamilton–Jacobi equations. As in part 1, we introduce by the convex duality Mather measures and their analogues for the system, which we call respectively Mather and Green–Poisson measures, and

In this paper we introduce Peierls–Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy \(\Gamma \)-converges to a limit functional comprised of two contributions: one is given by a constant \(c_\infty >0\) gauging the minimal energy

Necessary and sufficient conditions for rigidity of the perimeter inequality under spherical symmetrisation are given. That is, a characterisation for the uniqueness (up to orthogonal transformations) of the extremals is provided. This is obtained through a careful analysis of the equality cases, and studying fine properties of the circular symmetrisation, which was firstly introduced by Pólya in 1950

We study variational problems involving nonlocal supremal functionals $$\begin{aligned} L^\infty (\Omega ;{\mathbb {R}}^m) \ni u\mapsto \mathrm{ess sup}_{(x,y)\in \Omega \times \Omega } W(u(x), u(y)), \end{aligned}$$ where \(\Omega \subset \mathbb {R}^n\) is a bounded, open set and \(W:\mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}\) is a suitable function. Motivated by existence theory via

We give an existence result for first order evolution equation of the type \(\mathcal {R}u' + \mathcal {A}u = f\) where \(\mathcal {R}\) may be a function depending also on time assuming positive, null and negative sign, then the equation may be elliptic–parabolic, both forward and backward. The result is given in an abstract setting with Banach spaces depending on time (the functions u are defined

In this paper, we consider the following elliptic Toda system associated to any general simple Lie algebra with multiple singular sources $$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta w_i=\sum _{j=1}^na_{i,j}e^{2w_j}+2\pi \sum _{\ell =1}^m\beta _{i,\ell }\delta _{p_\ell } \quad &{}\hbox {in}\quad \mathbb {R}^2,\\ w_i(x)=-\,2\log |x|+O(1)~\hbox {as}~|x|\rightarrow \infty ,\quad &{} i=1,\ldots

We study the Dirichlet problem for the following prescribed mean curvature PDE $$\begin{aligned} {\left\{ \begin{array}{ll} -{\text {div}}\dfrac{\nabla v}{\sqrt{1+|\nabla v|^{2}}}=f(x,v) \text { in }\Omega \\ v=\varphi \text { on }\partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a domain contained in a complete Riemannian manifold M, \(f:\Omega \times \mathbb {R\rightarrow

By works of Schoen–Yau and Gromov–Lawson any Riemannian manifold with nonnegative scalar curvature and diffeomorphic to a torus is isometric to a flat torus. Gromov conjectured subconvergence of tori with respect to a weak Sobolev type metric when the scalar curvature goes to 0. We prove flat and intrinsic flat subconvergence to a flat torus for noncollapsing sequences of 3-dimensional tori \(M_j\)

We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Šešum and X.-J. Wang.

A diffusive Lotka–Volterra competition model is considered and the combined effect of spatial dispersal and spatial variations of resource on the population persistence and exclusion is studied. A new Lyapunov functional method and a new integral inequality are developed to prove the global stability of non-constant equilibrium solutions in heterogeneous environment. The general result is applied to

In this paper, we consider the asymptotic behavior of positive solutions of the biharmonic equation $$\begin{aligned} \Delta ^2 u = u^p\quad \text{ in }\; B_1\backslash \{0\} \end{aligned}$$ with an isolated singularity, where the punctured ball \(B_1 \backslash \{0\} \subset {\mathbb {R}}^n\) with \(n\ge 5\) and \(\frac{n}{n-4}< p < \frac{n+4}{n-4}\). This equation is relevant for the Q-curvature

We investigate regularity estimates for the stationary Navier–Stokes equations above a highly oscillating Lipschitz boundary with the no-slip boundary condition. Our main result is an improved Lipschitz regularity estimate at scales larger than the boundary layer thickness. We also obtain an improved \(C^{1,\mu }\) estimate and identify the building blocks of the regularity theory, dubbed ‘Navier polynomials’

Minimization problem on the action given by the Lagrangean is studied on the tangent bundle over a real Hilbert space. The existence of minimizers is proved by a compactness argument. The uniqueness of classical paths is proved under quadratic potentials.

We consider the following equation $$\begin{aligned} -\Delta u+u=Q_n(x)|u|^{p-2}u, \quad x\in {\mathbb {R}}^{N}, \end{aligned}$$ where \(Q_n\) are concrete bounded functions whose self-focusing core \(\text{ supp }\, Q_n^+\) shrinks to a finite set of points as \(n\rightarrow \infty \). We investigate the limiting profile of concentration for the ground state solutions and construct localized bound

We study blow-up and quantization phenomena for a sequence of solutions \((u_k)\) to the prescribed Q-curvature problem $$\begin{aligned} (-\Delta )^nu_k= Q_ke^{2nu_k}\quad \text {in }\Omega \subset {\mathbb {R}}^{2n},\quad \int _{\Omega }e^{2nu_k}dx\le C, \end{aligned}$$ under natural assumptions on \(Q_k\). It is well-known that, up to a subsequence, either \((u_k)\) is bounded in a suitable norm

In this paper, we discuss various comparison results between Laplacian (or bilaplacian) eigenvalues for a mean-convex bounded domain in a Riemannian manifold of non-negative Ricci curvature, under different boundary conditions, namely Dirichlet, Neumann, clamped and buckling conditions. We first obtain a new integral gradient estimate for a first Dirichlet eigenfunction, and deduce therefrom that the

This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime \({\mathbb {R}}^{1+3}\). We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime \({\mathbb {R}}^{1+3}\), the geometry of them are two spheres. The linear mode unstable of those lightlike

We prove some differentiable sphere theorems and topological sphere theorems for Lagrangian submanifolds in Kähler manifold and Legendrian submanifolds in Sasaki space form.

This paper is concerned with nonuniqueness and instability of the initial-boundary value problem for certain general systems of nonlinear diffusion equations. We explore the diffusion problems using the convex integration framework of nonhomogeneous space-time partial differential inclusions. Under a non-degeneracy (openness) condition called Condition (OC), we establish some nonuniqueness and instability

We propose a variational scheme for the construction of isentropic processes of the equations of adiabatic thermoelasticity with polyconvex internal energy. The scheme hinges on the embedding of the equations of adiabatic polyconvex thermoelasticity into a symmetrizable hyperbolic system. We establish existence of minimizers for an associated minimization theorem and construct measure-valued solutions