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

On a closed Riemannian manifold \((M,g_0)\) of even dimension \(n \ge 4\), the well-known prescribed Q-curvature problem asks whether there is a metric g comformal to \(g_0\) such that its Q-curvature, associated with the GJMS operator \(\mathbf {P}_g\), is equal to a given function f. Letting \(g = e^{2u}g_0\), this problem is equivalent to solving $$\begin{aligned} \mathbf {P}_{g_0} u+Q_{g_0} = f

We obtain an a-priori \(W_{{\mathrm{loc}}}^{1,\infty }\left( \Omega ; {\mathbb {R}}^{m}\right) \)-bound for weak solutions to the elliptic system $$\begin{aligned} \text {div}A\left( x,Du\right) =\sum _{i=1}^{n}\frac{\partial }{\partial x_{i}} a_{i}^{\alpha }\left( x,Du\right) =0,\;\;\;\;\;\alpha =1,2,\ldots ,m, \end{aligned}$$ where \(\Omega \) is an open set of \({\mathbb {R}}^{n}\), \(n\ge 2\),

In this paper we develop a comprehensive study on principal eigenvalues and maximum and comparison principles related to the nonlocal Lane–Emden problem $$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta )^{s}u = \lambda \rho (x)\vert v\vert ^{\alpha -1}v & \mathrm{in} \ \ \Omega ,\\ (-\Delta )^{t}v = \mu \tau (x)\vert u\vert ^{\beta -1}u & \mathrm{in} \ \ \Omega ,\\ u= v=0 & \mathrm{in} \ \ {\mathbb

We study a scalar elliptic problem in the data driven context. Our interest is to study the relaxation of a data set that consists of the union of a linear relation and single outlier. The data driven relaxation is given by the union of the linear relation and a truncated cone that connects the outlier with the linear subspace.

We introduce the transportation-annihilation distance\(W_p^\sharp \) between subprobabilities and derive contraction estimates with respect to this distance for the heat flow with homogeneous Dirichlet boundary conditions on an open set in a metric measure space. We also deduce the Bochner inequality for the Dirichlet Laplacian as well as gradient estimates for the associated Dirichlet heat flow. For

We establish the interior Hölder continuity for locally bounded solutions, and the Harnack inequality for non-negative continuous solutions to a class of anisotropic elliptic equations with bounded and measurable coefficients, whose prototype equation is $$\begin{aligned} u_{xx}+\varDelta _{q,y} u=0\quad {\text { locally in }}{\mathbb {R}}\times {\mathbb {R}}^{N-1},\quad {\text {for }}q<2, \end{aligned}$$

In this paper, we study the following nonlinear magnetic Schrödinger equation $$\begin{aligned} \left\{ \begin{aligned}&\Big (\frac{\varepsilon }{i}\nabla -A(x)\Big )^{2}u+V(x)u=f(|u|^{2})u \quad \hbox {in }{\mathbb {R}}^N\ (N\ge 2),\\&u\in H^{1}({\mathbb {R}}^{N}, {\mathbb {C}}), \end{aligned} \right. \end{aligned}$$ where \(\epsilon \) is a positive parameter, and \(V:{\mathbb {R}}^{N}\rightarrow

We consider the nonlocal Hénon equation $$\begin{aligned} (-\Delta )^s u= |x|^{\alpha } u^{p},\quad {\mathbb {R}}^{N}, \end{aligned}$$ where \((-\Delta )^s\) is the fractional Laplacian operator with \(01\) and \(N>2s\). We prove a nonexistence result for positive solutions in the optimal range of the nonlinearity, that is, when $$\begin{aligned} 1

We show that the Hamiltonian action satisfies the Palais–Smale condition over a “mixed regularity” space of loops in cotangent bundles, namely the space of loops with regularity \(H^s\), \(s\in (\frac{1}{2}, 1)\), in the base and \(H^{1-s}\) in the fiber direction. As an application, we give a simplified proof of a theorem of Hofer–Viterbo on the existence of closed characteristic leaves for certain

We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first

We prove a generalized Bôcher-type theorem for the nonhomogeneous Laplace’s equations on singular manifolds with conical metrics. More specifically, we give a sharp characterization of the behavior at isolated singularities of a solution bounded on one side for the equation \(\Delta _g u =f\) (\(f \in L_g^q\) with \(q > \frac{n}{2}\)). The main results in this paper imply that a nonnegative solution

In this paper, we study the three-dimensional non-isentropic compressible fluid–particle flows. The system involves coupling between the Vlasov–Fokker–Planck equation and the non-isentropic compressible Navier–Stokes equations through momentum and energy exchanges. For the initial data near the given equilibrium we prove the global well-posedness of strong solutions and obtain the optimal algebraic

Let \(\varOmega \) be a bounded open domain in \(\mathbb {R}^n\) and \(\triangle _{\mu }=\sum _{j=1}^{n}\partial _{x_{j}}(\mu _{j}^2(x)\partial _{x_{j}})\) be a class of degenerate elliptic operator with continuous nonnegative coefficients \(\mu _{1},\mu _{2},\ldots ,\mu _{n}\). Denote by \(\lambda _{k}\) the kth Dirichlet eigenvalue of the self-adjoint degenerate elliptic operator \(-\triangle _{\mu

In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We detect the product of two species masses as a feature to determine boundedness, gradient estimate, blow-up and exponential convergence of classical

The purpose of this paper is to show that in elastoplastic materials cracks can grow only in an intermittent way. This result is rigorously proved in the framework of a simplified model.

We provide a general condition on the kernel of an integro-differential operator so that its associated quadratic form satisfies a coercivity estimate with respect to the \(H^s\)-seminorm.

We consider in this work the problem of minimizing the von Neumann entropy under the constraints that the density of particles, the current, and the kinetic energy of the system is fixed at each point of space. The unique minimizer is a self-adjoint positive trace class operator, and our objective is to characterize its form. We will show that this minimizer is solution to a self-consistent nonlinear

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry–Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class \(\mathbf {B}^{1,1/2}(X)\) that was introduced in our previous paper. Assuming furthermore a quasi Bakry–Émery

Let \(\Omega \subseteq {\mathbb {R}}^{d}\) be open and A a complex uniformly strictly accretive \(d\times d\) matrix-valued function on \(\Omega \) with \(L^{\infty }\) coefficients. Consider the divergence-form operator \({{\mathscr {L}}}^{A}=-\mathrm{div}\,(A\nabla )\) with mixed boundary conditions on \(\Omega \). We extend the bilinear inequality that we proved in Carbonaro and Dragičević (J Eur

Let \(\Omega ^+\subset {\mathbb {R}}^{n+1}\) be an NTA domain and let \(\Omega ^-= {\mathbb {R}}^{n+1}\setminus \overline{\Omega ^+}\) be an NTA domain as well. Denote by \(\omega ^+\) and \(\omega ^-\) their respective harmonic measures. Assume that \(\Omega ^+\) is a \(\delta \)-Reifenberg flat domain for some \(\delta >0\) small enough. In this paper we show that \(\log \frac{d\omega ^-}{d\omega

We give criteria for the existence of bifurcations of symmetric periodic orbits in reversible Hamiltonian systems in terms of local equivariant Lagrangian Rabinowitz Floer homology. As an example, we consider the family of the direct circular orbits in the rotating Kepler problem and observe bifurcations of torus-type orbits. Our setup is motivated by numerical work of Hénon on Hill’s lunar problem

The solvability in Sobolev spaces with special mixed norms is proved for nondivergence form second order parabolic equations. The usefulness of mixed norms in our problem is similar to that in the theory of Navier–Stokes equation, where mixed norms are indispensable and quite powerful. We obtain better regularity properties of solutions which allow us to apply the results to proving weak uniqueness

We study a concentration and homogenization problem modelling electrical conduction in a composite material. The novelty of the problem is due to the specific scaling of the physical quantities characterizing the dielectric component of the composite. This leads to the appearance of a peculiar displacement current governed by a Laplace–Beltrami pseudo-parabolic equation. This pseudo-parabolic character

Let \(\Omega \subset \mathbb {R}^{N}\), \(N \ge 2\), be a smooth bounded domain. We consider the boundary value problem where \(c_{\lambda }\) and h belong to \(L^q(\Omega )\) for some \(q > N/2\), \(\mu \) belongs to \(\mathbb {R}{\setminus } \{0\}\) and we write \(c_{\lambda }\) under the form \(c_{\lambda }:= \lambda c_{+} - c_{-}\) with \(c_{+} \gneqq 0\), \(c_{-} \ge 0\), \(c_{+} c_{-} \equiv

In this paper we consider the quasilinear critical problem $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta _{p}u=\lambda \left| u\right| ^{q-2}u+\left| u\right| ^{p^{\star }-2}u &{} \mathrm {in}\;\Omega ,\\ u=0 &{} \mathrm {on}\;\partial \Omega , \end{array}\right. \end{aligned}$$ where \(\Omega \) is a bounded domain in \({\mathbb {R}}^{N}\) with smooth boundary, \(-\Delta _{p}u:=\mathrm

We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent variational characterization of Lipschitz transport map by the second author and Juillet with a convexity property of optimizers in the dual formulation of the entropy-regularized

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of

In this paper, we show that one-dimension systems of quasilinear wave equations with null conditions admit global classical solutions for small initial data. This result extends Luli, Yang and Yu’s seminal work (Luli et al. in Adv Math 329:174–188, 2018) from the semilinear case to the quasilinear case. Furthermore, we also prove that the global solution is asymptotically free in the energy sense.

We address the minimization of the Canham–Helfrich functional in presence of multiple phases. The problem is inspired by the modelization of heterogeneous biological membranes, which may feature variable bending rigidities and spontaneous curvatures. With respect to previous contributions, no symmetry of the minimizers is here assumed. Correspondingly, the problem is reformulated and solved in the

We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does not enjoy a gauge invariance. The uniqueness result is complemented by an associated

Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity \((-\Delta )^{\theta }\), whenever the exponent \(\theta \) is less than Lions’ exponent 5/4, i.e., when \(\theta < 5/4\).

In this paper we study theoretical properties of the entropy-transport functional with repulsive cost functions. We provide sufficient conditions for the existence of a minimizer in a class of metric spaces and prove the \(\Gamma \)-convergence of the entropy-transport functional to a multi-marginal optimal transport problem with a repulsive cost. We point out that our construction can deal with the

We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality $$\begin{aligned} \Vert e^{-\tau \ell (\cdot )} u^{\frac{1}{q}} f\Vert _q\le c_\tau \Vert e^{-\tau \ell (\cdot )} v^{\frac{1}{p}}\, \nabla f\Vert _p, \quad f\in C^\infty _0({\mathbb {R}}^n). \end{aligned}$$ This inequality is a Carleman-type estimate that yields unique continuation results for solutions

In this paper we deal with a doubly nonlinear Cahn–Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential includes two regularizations: a viscous and a diffusive term. First of all, we prove existence and uniqueness of a bounded solution to the system using a nonstandard

It is a consequence of the Morse–Bott Lemma (see Theorems 2.10 and 2.14) that a \(C^2\) Morse–Bott function on an open neighborhood of a critical point in a Banach space obeys a Łojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses (Theorems 1, 2, and Corollary 3) for analytic functions on Banach spaces: If the Łojasiewicz exponent of an analytic function

We use non-variational type arguments to prove optimal regularity and smoothness of the free boundary for one-phase solutions to inhomogeneous nonlinear free boundary problems (FBP) governed by singular/degenerate elliptic PDEs with a nonzero right hand side (RHS). In a precise way, we show that viscosity solutions to FBP, as previously mentioned, are locally Lipschitz continuous and under certain

We prove that for elliptic and canceling linear partial differential operators \({\mathbb {B}}\) of order n on \({\mathbb {R}}^n\), continuity of a map u can be inferred from the fact that \({\mathbb {B}}u\) is a measure. We also prove strict continuity of the embedding of the space \({\text {BV}}^{\mathbb {B}}({\mathbb {R}}^n)\) into the space of continuous functions vanishing at infinity. Here, \({\text

Under the condition of small external forces, we obtain existence of a weak solution of the steady Hall-MHD system with Hölder continuous magnetic field. We also established regularity of weak solutions provided that magnetic fields are bounded. For sufficiently small external forces, uniqueness result is also established.

The paper offers a self-contained account of the theory of first and second order necessary conditions for optimal control problems (with state constraints) based on new principles coming from variational analysis. The key element of the theory is reduction of the problem to unconstrained minimization of a Bolza-type functional with necessarily non-differentiable integrand and off-integral term. This

We consider a quasi-variational inequality governed by a moving set. We employ the assumption that the movement of the set has a small Lipschitz constant. Under this requirement, we show that the quasi-variational inequality has a unique solution which depends Lipschitz-continuously on the source term. If the data of the problem is (directionally) differentiable, the solution map is directionally differentiable

We study a linear parabolic equation, involving inverse square potential, defined on the whole space. It turns out, that the asymptotic behavior of its solution, is not a decay to the trivial function. For this, we prove new results for the Hardy inequality on \({\mathbb {R}}^N\).

In this paper, we will study the local uniqueness of single bubbling solutions for elliptic problems of Ambrosetti–Prodi type with critical growth in a unit ball. As a result, we are able to count the exact number of single bubbling solutions for these type of elliptic problems.

We study the evolution of star-shaped sets in volume preserving mean curvature flow. Constructed by approximate minimizing movements, our solution preserves a strong version of star-shapedness. We also show that the solution converges to a ball as time goes to infinity. For asymptotic behavior of the solution we use the gradient flow structure of the problem, whereas a modified notion of viscosity

We construct, for a locally compact metric space X, a space of polylipschitz forms \({\overline{\Gamma }}^*_c(X)\), which is a pre-dual for the space of metric currents \({\mathscr {D}}_*(X)\) of Ambrosio and Kirchheim. These polylipschitz forms may be seen as an analog of differential forms in the metric setting.

Given a bounded domain of \({{\mathbb {R}}}^{n}\) of class \( C^{2}\), we prove the symmetry of solutions to overdetermined problems obtained by adding both zero Dirichlet and constant Neumann boundary conditions to a class of fully nonlinear equations \(\sigma _{k}(\lambda )=C_{n}^{k}\), where \(\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _n)\) are the principal curvatures of a graph. Our method

In this paper, we study the convergence results for the harmonic maps between Alexandrov spaces under the Gromov–Hausdroff convergence. We prove that the harmonicity of a map is presevered under Gromov–Hausdroff convergence with the noncollapsing assumption.

The paper is devoted to the classification of entire solutions to the Cahn–Hilliard equation \(-\Delta u=u-u^3-\delta \) in \(\mathbb {R}^N\), with particular interest in those solutions whose nodal set is either bounded or contained in a cylinder. The aim is to prove either radial or cylindrical symmetry, under suitable hypothesis.

We study the question whether Lipschitz minimizers of \(\int F(\nabla u)\,dx\) in \(\mathbb {R}^n\) are \(C^1\) when F is strictly convex. Building on work of De Silva–Savin, we confirm the \(C^1\) regularity when \(D^2F\) is positive and bounded away from finitely many points that lie in a 2-plane. We then construct a counterexample in \(\mathbb {R}^4\), where F is strictly convex but \(D^2F\) degenerates

We study measures \(\mu \) on the plane with two independent Alberti representations. It is known, due to Alberti, Csörnyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A–C–P. Assuming that the representations of \(\mu \) are bounded from above, in a natural way to be defined in the introduction, we

We consider a point cloud \(X_n := \{ {\mathbf {x}}_1, \ldots , {\mathbf {x}}_n \}\) uniformly distributed on the flat torus \({\mathbb {T}}^d : = \mathbb {R}^d / \mathbb {Z}^d \), and construct a geometric graph on the cloud by connecting points that are within distance \(\varepsilon \) of each other. We let \({\mathcal {P}}(X_n)\) be the space of probability measures on \(X_n\) and endow it with

We consider the Robin boundary value problem \({\mathrm {div}}\,(A\nabla u) = {\mathrm {div}}\,\varvec{f}+F\) in \(\Omega \), a \(C^1\) domain, with \((A\nabla u - \varvec{f})\cdot {\varvec{n}}+ \alpha u = g\) on \(\Gamma \), where the matrix A belongs to \(VMO ({\mathbb {R}}^3) \), and discover the uniform estimates on \(\Vert u\Vert _{W^{1,p}(\Omega )}\), with \(1< p < \infty \), independent of \(\alpha

We consider the boundary value problem $$\begin{aligned} \begin{array}{rcl} -\Delta _p u &{} = &{} \alpha |u|^{p-2}u^+-\beta |u|^{p-2}u^- \text{ in } \Omega ,\\ u &{} = &{} 0 \text{ on } \partial \Omega , \end{array} \end{aligned}$$ where \(\Delta _p u:=\nabla \cdot (|\nabla u|^{p-2}\nabla u)\) for \(1

Given \(\alpha >0\), we establish the following two supercritical Moser–Trudinger inequalities $$\begin{aligned} \mathop {\sup }\limits _{ u \in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla u|^n dx \le 1 } \int _B \exp \big ( (\alpha _n + |x|^\alpha ) |u|^{\frac{n}{n-1}} \big ) dx < +\infty \end{aligned}$$ and $$\begin{aligned} \mathop {\sup }\limits _{ u\in W^{1,n}_{0,\mathrm{rad}}(B): \int _B |\nabla

In this paper, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter \(\varepsilon \), which is singular for the flows, are established via a variational approach based on the compressible–incompressible

Abstract We consider nonlocal variational problems in \(L^p\), like those that appear in peridynamics, where the functional object of the study is given by a double integral. It is known that convexity of the integrand implies the lower semicontinuity of the functional in the weak topology of \(L^p\). If the integrand is not convex, a usual approach is to compute the relaxation, which is the lower

Abstract Nonexistence of nontrivial nonnegative classical solutions is obtained for the problems: 0.1$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2 u=u^p \;\;\; &{}\text{ in } {\mathbb {R}}^N \backslash {\overline{B}},\\ u=\Delta u=0 \;\;\; &{}\text{ on } \partial B \end{array} \right. \end{aligned}$$with \(1\frac{N+4}{N-4}\) respectively.

Abstract We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak \((1,1)\)-Poincaré inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional

Abstract We consider the inverse multiphase Stefan problem with homogeneous Dirichlet boundary condition on a bounded Lipschitz domain, where the density of the heat source is unknown in addition to the temperature and the phase transition boundaries. The variational formulation is pursued in the optimal control framework, where the density of the heat source is a control parameter, and the criteria

Abstract Many models in mathematical physics are given as non-linear partial differential equation of hydrodynamic type; the incompressible Euler, KdV, and Camassa–Holm equations are well-studied examples. A beautiful approach to well-posedness is to go from the Eulerian to a Lagrangian description. Geometrically it corresponds to a geodesic initial value problem on the infinite-dimensional group of