In this paper, we define a new discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study

We study the line bundle mean curvature flow on Kähler surfaces under the hypercritical phase and a certain semipositivity condition. We naturally encounter such a condition when considering the blowup of Kähler surfaces. We show that the flow converges smoothly to a singular solution to the deformed Hermitian–Yang–Mills equation away from a finite number of curves of negative self-intersection on

The present paper deals with a parametrized Kirchhoff type problem involving a critical nonlinearity in high dimension. Existence, non existence and multiplicity of solutions are obtained under the effect of a subcritical perturbation by combining variational properties with a careful analysis of the fiber maps of the energy functional associated to the problem. The particular case of a pure power

We study the higher Hölder regularity of local weak solutions to a class of nonlinear nonlocal elliptic equations with kernels that satisfy a mild continuity assumption. An interesting feature of our main result is that the obtained regularity is better than one might expect when considering corresponding results for local elliptic equations in divergence form with continuous coefficients. Therefore

In this paper we show the uniqueness of the critical point for semi-stable solutions of the problem $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=f(u)&{}\quad \text {in }\Omega \\ u>0&{}\quad \text {in }\Omega \\ u=0&{}\quad \text {on }\partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \subset \mathbb {R}^2\) is a smooth bounded domain whose boundary has nonnegative curvature

The central object of this article is (a special version of) the Helfrich functional which is the sum of the Willmore functional and the area functional times a weight factor \(\varepsilon \ge 0\). We collect several results concerning the existence of solutions to a Dirichlet boundary value problem for Helfrich surfaces of revolution and cover some specific regimes of boundary conditions and weight

In this paper we give local curvature estimates for the Laplacian flow on closed \(G_{2}\)-structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s

In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in \(H^{s}\) with spatial dimension \(n \le 5\). We show this equation, with power \(2\le p\le 1+4/(n-1)\), is (strongly) ill-posed in \(H^{s}\) with \(s = (n+5)/4\) in general. Moreover, when the nonlinearity is quadratic we establish a characterization of the structure

In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive–attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the \(\lambda \)-Wasserstein metric topology with \(1\le \lambda <\infty \), provided masses and distance of Dirac deltas are equally half and one, respectively. In addition

We consider degenerate and singular parabolic equations with p-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure

In this work we consider the homogeneous Neumann eigenvalue problem for the Laplacian on a bounded Lipschitz domain and a singular perturbation of it, which consists in prescribing zero Dirichlet boundary conditions on a small subset of the boundary. We first describe the sharp asymptotic behaviour of a perturbed eigenvalue, in the case in which it is converging to a simple eigenvalue of the limit

In this paper, we study the deformed Hermitian-Yang-Mills equation on compact Kähler manifold with non-negative orthogonal bisectional curvature. We prove that the curvatures of deformed Hermitian-Yang-Mills metrics are parallel with respect to the background metric if there exists a positive constant C such that $$\begin{aligned} -\frac{1}{C}\omega<\sqrt{-1}F

In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give its applications to the Hessian equations, Hessian quotient equations and the special

We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker (J Differ Geom 85(3):357–395, 2010) have shown that initial submanifolds satisfying this pinching

We present a self-contained and comprehensive study of the Fisher-Rao space of matrix-valued non-commutative probability measures, and of the related Hellinger space. Our non-commutative Fisher-Rao space is a natural generalization of the classical commutative Fisher-Rao space of probability measures and of the Bures-Wasserstein space of Hermitian positive-definite matrices. We introduce and justify

In this paper, we consider the Dirichlet problem of a complex Monge–Ampère equation on a ball in \({\mathbb {C}}^n\). With \({\mathcal {C}}^{1,\alpha }\) (resp. \({\mathcal {C}}^{0,\alpha }\)) data, we prove an interior \({\mathcal {C}}^{1,\alpha }\) (resp. \({\mathcal {C}}^{0,\alpha }\)) estimate for the solution. These estimates are generalized versions of the Bedford–Taylor interior \({\mathcal

We consider stable solutions of a semilinear elliptic equation with homogeneous Neumann boundary conditions. A classical result of Casten, Holland and Matano states that all stable solutions are constant in convex bounded domains. In this paper, we examine whether this result extends to unbounded convex domains. We give a positive answer for stable non-degenerate solutions, and for stable solutions

In this paper we deal with large solutions to $$\begin{aligned} {\left\{ \begin{array}{ll} u - \Delta _{p} u + \beta |\nabla u|^{q} =f&{} \text{ in } \,\Omega ,\\ u (x) = +\infty &{} \text{ on } \,\partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \subset {\mathbb {R}}^N\) , with \(N\ge 1\), is a smooth, open, connected, and bounded domain, \(p \ge 2\), \(\beta >0\), \(p-1

In this paper we present a detailed study of critical embeddings of weighted Sobolev spaces into weighted Orlicz spaces of exponential type for weights of monomial type. More precisely, we give an alternative proof of a recent result by N. Lam [NoDEA 24(4), 2017] showing the optimality of the constant in the Trudinger–Moser inequality. We prove a Poincaré inequality for this class of weights. We show

We consider the n-dimensional (\(n=2,3\)) Camassa–Holm equations with nonlocal diffusion of type \((-\,\Delta )^{s}, \ \frac{n}{4}\le s<1\). In Gan et al. (Discrete Contin Dyn Syst 40(6):3427–3450, 2020), the global-in-time existence and uniqueness of finite energy weak solutions is established. In this paper, we show that with regular initial data, the finite energy weak solutions are indeed regular

In this paper, we consider the existence of optimizers for the following Sobolev and Gagliardo–Nirenberg intepolation inequalities in \( \dot{W}^{s,p}({\mathbb {R}}^d) \) at the non-endpoint case: $$\begin{aligned} ||u||_{L^{p^*}} \le C || u||_{ \dot{W}^{s,p}}, \qquad ||u||_{L^r} \le C || u||_{\dot{W}^{s_1,p}}^{\theta } ||u||_{L^p}^{1-\theta }, \end{aligned}$$ where \( \dot{W}^{s,p} ({\mathbb {R}}^d)

We investigate a strong maximum principle of Vázquez type for viscosity solutions of fully nonlinear and degenerate elliptic equations involving Hörmander vector fields. We also give a strong comparison principle for such equations.

We prove existence results in all of \({\mathbb {R}}^N\) for an elliptic problem of (p, q)-Laplacian type involving a critical term, nonnegative weights and a positive parameter \(\lambda \). In particular, under suitable conditions on the exponents of the nonlinearity, we prove existence of infinitely many weak solutions with negative energy when \(\lambda \) belongs to a certain interval. Our proofs

In this paper we study localization properties of the Riesz s-fractional gradient \(D^s u\) of a vectorial function u as \(s \nearrow 1\). The natural space to work with s-fractional gradients is the Bessel space \(H^{s,p}\) for \(0< s < 1\) and \(1< p < \infty \). This space converges, in a precise sense, to the Sobolev space \(W^{1,p}\) when \(s \nearrow 1\). We prove that the s-fractional gradient

The Special Lagrangian Potential Equation for a function u on a domain \(\Omega \subset {{\mathbb {R}}}^n\) is given by \({\mathrm{tr}}\{\arctan (D^2 \,u) \} = \theta \) for a contant \(\theta \in (-n {\pi \over 2}, n {\pi \over 2})\). For \(C^2\) solutions the graph of Du in \(\Omega \times {{\mathbb {R}}}^n\) is a special Lagrangian submanfold. Much has been understood about the Dirichlet problem

In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is \(C^{1/2}\)-Hölder continuous. The coefficient of the damping depends on time, given by this form \(\frac{\mu }{(1+t)^\lambda }\), \(\lambda ,\ \mu >0\), which decays by order \(-\lambda \) in time

We show that the N-covering map, which in complex coordinates is given by \(u_{_{\scriptscriptstyle {N}}}(z):=z \mapsto z^{N}/\sqrt{N}|z|^{N-1}\) and where N is a natural number, is a global minimizer of the Dirichlet energy \(\mathbb {D}(v)=\int _B |\nabla v(x)|^2 \, dx\) with respect to so-called inner and outer variations. An inner variation of \(u_{_{\scriptscriptstyle {N}}}\) is a map of the form

We prove index estimates for closed and free boundary CMC surfaces in certain 3-dimensional submanifolds of some Euclidean space. When the mean curvature is large enough we are able to prove that the index of a CMC surface in an arbitrary 3-manifold is bounded below by a linear function of its genus.

We consider the minimization of an energy functional given by the sum of a density perimeter and a nonlocal interaction of Riesz type with exponent \(\alpha \), under volume constraint, where the strength of the nonlocal interaction is controlled by a parameter \(\gamma \). We show that for a wide class of density functions the energy admits a minimizer for any value of \(\gamma \). Moreover these

We investigate Lamé systems in periodically perforated domains, and establish quantitative homogenization results in the setting where the domain is clamped at the boundary of the holes. Our method is based on layer potentials and it provides a unified proof for various regimes of hole-cell ratios (the ratio between the size of the holes and the size of the periodic cells), and, more importantly, it

We prove strong convergence of the Bopp–Podolsky–Schrödinger–Proca system to the Schrödinger–Poisson–Proca system in the electro-magneto-static case as the Bopp–Podolsky parameter goes to zero.

We introduce a new method for constructing solenoidal extensions of fairly general boundary data in (2d or 3d) cubes that contain an obstacle. This method allows us to provide explicit bounds for the Dirichlet norm of the extensions. It runs as follows: by inverting the trace operator, we first determine suitable extensions, not necessarily solenoidal, of the data; then we analyze the Bogovskii problem

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form \(\Delta u+g(u)=0\). Our result applies in particular to the double power non-linearity where \(g(u)=u^q-u^p-\mu u\) for \(p>q>1\) and \(\mu >0\), which we discuss with more details. In this case, the non-degeneracy of the unique solution \(u_\mu \) allows us to

We establish uniqueness of vanishing radially decreasing entire solutions, which we call ground states, to some semilinear fractional elliptic equations. In particular, we treat the fractional plasma equation and the supercritical power nonlinearity. As an application, we deduce uniqueness of radial steady states for nonlocal aggregation-diffusion equations of Keller-Segel type, even in the regime

In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estimate, the no-local-collapsing theorem, the pseudo-locality theorem and the strong

We analyze a finite-difference approximation of a functional of Ambrosio–Tortorelli type in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step \(\delta \) is smaller than the ellipticity parameter \(\varepsilon \), we show the \(\varGamma \)-convergence of the model to the Griffith functional, containing only a

We show that closed, immersed, minimal hypersurfaces in a compact symmetric space satisfy a lower bound on the index plus nullity, which depends linearly on their first Betti number. Moreover, if either the minimal hypersurface satisfies a certain genericity condition, or if the ambient space is a product of two CROSSes, we improve this to a lower bound on the index alone, which is affine in the first

For the spatially inhomogeneous, non-cutoff Boltzmann equation posed in the whole space \({\mathbb {R}}^3_x\), we establish pointwise lower bounds that appear instantaneously even if the initial data contains vacuum regions. Our lower bounds depend only on the initial data and upper bounds for the mass and energy densities of the solution. As an application, we improve the weakest known continuation

We establish the subconvergence of weak solutions to the Ginzburg–Landau approximation to global-in-time weak solutions of the Ericksen–Leslie model for nematic liquid crystals on the torus \({\mathbb {T}^2}\). The key argument is a variation of concentration-cancellation methods originally introduced by DiPerna and Majda to investigate the weak stability of solutions to the (steady-state) Euler equations

We prove a global weighted \(L^{p(\cdot )}\)-regularity for the Hessian of strong solution to the Cauchy–Dirichlet problem for fully nonlinear parabolic equations in a bounded \(C^{1,1}\)-domain, where the associated nonlinearity is \((\delta ,R)\)-vanishing in independent variables, the variable exponent \(p(\cdot )\) is \(\log \)-Hölder continuous, and the weight \(\omega \) is of the \(A_{p(\cdot

In this work, we prove the existence of a third embedded minimal hypersurface spanning a closed submanifold \(\gamma \), of mountain pass type, contained in the boundary of a compact Riemannian manifold with convex boundary, when it is known a priori the existence of two strictly stable minimal hypersurfaces that bound \(\gamma \). In order to do so, we develop min–max methods similar to those of De

Let \({\mathcal {H}}_{\alpha }=\Delta -(\alpha -1)|x|^{\alpha }\) be an \([1,\infty )\ni \alpha \)-Hermite operator for the hydrogen atom located at the origin in \({\mathbb {R}}^d\). In this paper, we are motivated by the classical case \(\alpha =1\) to investigate the space of functions with \(\alpha \)-Hermite Bounded Variation and its functional capacity and geometrical perimeter.

We study the limiting behavior as \(p\uparrow 2\) of the singular sets \(Sing(u_p)\) and p-energy measures \(\mu _p:=(2-p)|du_p|^pdvol\) for families of stationary p-harmonic maps \(u_p\in W^{1,p}(M,S^1)\) from a closed, oriented manifold M to the circle. When the measures \(\mu _p\) have uniformly bounded mass, we show that—up to subsequences—the singular sets \(Sing(u_p)\) converge in the Hausdorff

In this paper, we first give a necessary and sufficient condition for the boundedness and the compactness of a class of nonlinear functionals in \(H^{2}\left( {\mathbb {R}}^{4}\right) \) which are of their independent interests. (See Theorems 2.1 and 2.2.) Using this result and the principle of symmetric criticality, we can present a relationship between the existence of the nontrivial solutions to

We study the asymptotic limit of diffused surface energy in the van der Waals–Cahn–Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min–max method.

An optimization problem with volume constraint involving the \(\varPhi \)-Laplacian in Orlicz–Sobolev spaces is considered for the case where \(\varPhi \) does not satisfy the natural condition introduced by Lieberman. A minimizer \(u_\varPhi \) having non-degeneracy at the free boundary is proved to exist and some important consequences are established like the Lipschitz regularity of \(u_ \varPhi

We verify that if M is a compact minimal submanifold with boundary in a Cartan–Hadamard manifold N, then \(\sum \nolimits _{i=1}^{k}\lambda _i\ge \frac{2n\pi }{e}k^{\frac{n+2}{n}}{{\,\mathrm{vol}\,}}(M)^{-\frac{2}{n}}\), where \(\lambda _i\) is the ith Dirichlet eigenvalue of the Laplacian on M. This provides an evidence for that the generalized Pólya conjecture is true. We also prove that if M is

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 \(\Delta _{\mathbb {H}^{N}}\)