In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity.

In this paper, we establish uniform a priori estimates for positive solutions to higher critical order fractional Lame-Emden equations for all large exponents by adopting a new idea of using the combination of Green’s representations with suitable cut-off functions. We obtain two key ingredients needed in the estimates. One is the monotonicity of solutions in the inward normal direction near the boundary

This paper is concerned with the following Maxwell–Dirac system $$\begin{aligned}&-i\sum ^3_{k=1}\alpha _{k}\partial _{k}u + (V(x)+a)\beta u + \omega u-K(x)\phi u =F_u(x,u),\\&\quad -\Delta \phi =4\pi K(x)|u|^2, \end{aligned}$$ in \(\mathbb {R}^3\), where V(x) is a potential function and F(x, u) is a nonlinear function modeling various types of interaction and K(t, x) is the varying pointwise charge

We consider Hamilton–Jacobi equations in one space dimension with Hamiltonians of the form \(H(p,x,\omega ) = G(p) +\beta V(x,\omega )\), where \(V(\cdot ,\omega )\) is a stationary and ergodic potential of unit amplitude. The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive G. Under the extra condition that G is a double-well

We prove optimal gradient estimates for distributional solutions to non-uniformly elliptic equations of multi-phase type in divergence form by investigating sharp conditions on such nonlinear operators for the Calderón-Zygmund theory.

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the so-called Mean Field Game system with state constraints. For this, we show a global semiconvavity property of the value function associated with optimal control problems

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular

We study rays and co-rays in the Wasserstein space \(P_p({\mathcal {X}})\) (\(p > 1\)) whose ambient space \({\mathcal {X}}\) is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient space. We show the existence of co-rays for any prescribed initial probability

We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge–Ampère equation on a compact Hermitian manifold for a very general measure on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh–Nguyen–Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic

In the present paper, we develop a direct approach to find nontrivial solutions and ground state solutions for the following planar Schrödinger equation: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u=f(x,u), \;\;&{} x\in {{\mathbb {R}}}^{2},\\ u\in H^1({\mathbb {R}}^2), \end{array}\right. } \end{aligned}$$ where V(x) is an 1-periodic function with respect to \(x_1\) and \(x_2\), 0 lies

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than \(\varLambda _{n-2}\). In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption \(H - \frac{\langle x, \nu \rangle }{2} > 0\) is possible and as an application we use the surgery flow to show that smooth n-dimensional closed self

In this paper, we establish the curvature estimates for a class of Hessian type equations. Some applications are also discussed.

We consider the homogeneous Bose gas on a unit torus in the mean-field regime when the interaction strength is proportional to the inverse of the particle number. In the limit when the number of particles becomes large, we derive a two-term expansion of the one-body density matrix of the ground state. The proof is based on a cubic correction to Bogoliubov’s approximation of the ground state energy

We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight \(|y|^a\) for \(a \in (-1,1)\). Such problem arises as the local extension of the obstacle problem for the fractional heat operator \((\partial _t - \Delta _x)^s\) for \(s \in (0,1)\). Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve

In this paper, we show the existence of nontopological solutions to the self-dual Einstein-Maxwell-Higgs equations on a compact surface when the parameter \({\varepsilon }>0\) is small enough and the total string number N is bigger than two. This improves the previously known results by removing any extra conditions on N. Based on degree theory and blowup analysis, we introduce a new method to obtain

This paper is concerned with global existence of classical solutions as well as occurrence of infinite-time blowups to the following fully parabolic system $$\begin{aligned} {\left\{ \begin{array}{ll} u_t=\varDelta (\gamma (v)u)\\ v_t-\varDelta v+v=u \end{array}\right. } \end{aligned}$$(1) in a smooth bounded domain \(\varOmega \subset {\mathbb {R}}^n\), \(n\ge 1\) with no-flux boundary conditions

We study the Abels–Garcke–Grün (AGG) model for a mixture of two viscous incompressible fluids with different densities. The AGG model consists of a Navier–Stokes–Cahn–Hilliard system characterized by a (non-constant) concentration-dependent density and an additional flux term due to interface diffusion. In this paper we address the well-posedness problem in the two-dimensional case. We first prove

In this paper we study general transportation problems in \({\mathbb {R}}^n\), in which m different goods are moved simultaneously. The initial and final positions of the goods are prescribed by measures \(\mu ^-\), \(\mu ^+\) on \({\mathbb {R}}^n\) with values in \({\mathbb {R}}^m\). When the measures are finite atomic, a discrete transportation network is a measure T on \({\mathbb {R}}^n\) with values

We are concerned with the planar \(L_p\) dual Minkowski problem with indices p, q. Through the compactness analysis of an associated constrained variational problem in Sobolev space, the solvability of the planar \(L_p\) dual Minkowski problem and the related functional inequality are established, upon which the multiple solutions to the planar \(L_p\) dual Minkowski problem are obtained. Precisely

We consider divergence form, second-order strongly parabolic systems in a cylindrical domain with a finite number of subdomains under the assumption that the interfacial boundaries are \(C^{1,\text {Dini}}\) and \(C^{\gamma _{0}}\) in the spatial variables and the time variable, respectively. Gradient estimates and piecewise \(C^{1/2,1}\)-regularity are established when the leading coefficients and

We introduce and study a new optimal transport problem on a bounded domain \({{\bar{\Omega }}}\subset {\mathbb {R}}^d\), defined via a dynamical Benamou–Brenier formulation. The model handles differently the motion in the interior and on the boundary, and penalizes the transfer of mass between the two. The resulting distance interpolates between classical optimal transport on \({{\bar{\Omega }}}\)

We provide a general approach to deform framed curves while preserving their clamped boundary conditions (this includes closed framed curves) as well as properties of their curvatures. We apply this to director theories, which involve a curve \(\gamma : (0, 1)\rightarrow \mathbb {R}^3\) and orthonormal directors \(d_1\), \(d_2\), \(d_3: (0,1)\rightarrow \mathbb {S}^2\) with \(d_1 = \gamma '\). We show

In this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent papers (De Lellis et al. in Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335; Tione in Minimal graphs and differential inclusions. Commun Part Differ Equ 7:1–33

This paper deals with the semilinear elliptic problem $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda m(x)u-[a(x)+\varepsilon b(x)]u^p &{}\text { in } \Omega ,\\ Bu=0 &{}\text { on } \partial \Omega , \end{array}\right. } \end{aligned}$$ where \(p>1\), \(\lambda >0\), \(m,a,b\in C({\bar{\Omega }})\), with \(a \gneq 0\), \(b\gneq 0\), \(\Omega \) is a bounded \(C^{2}\) domain of \({\mathbb

We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.

In this note we give existence results for the generalized Tricomi equations \({\mathcal {R}}u'' + {\mathcal {B}}u = f\) and \(({\mathcal {R}}u')' + {\mathcal {B}}u = f\) with suitable boundary data where \({\mathcal {R}}\) may be an operator (or a function) depending also on time assuming positive, null and negative sign, while \({\mathcal {B}}\) is an elliptic operator. To do that we also extend

We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which has z-dependent zeros of constant sign. For all big values of the parameter \(\lambda >0\), we prove two multiplicity theorems producing two positive solutions, two negative solutions, and two nodal solutions. In the first we do not impose any asymptotic conditions on the reaction

For an immortal Ricci flow on an m-dimensional \((m\ge 3)\) closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a Riemannian orbifold; (2) if the flow is type-III with diameter growth controlled by \(t^{\frac{1}{2}}\), then any blowdown limit is an m-dimensional negative Einstein

We investigate the existence of distributional solutions to nonlinear elliptic problems with general growth when the right-hand side is a bounded Radon measure. An optimal global Calderón-Zygmund type estimate for such problems is obtained by using mapping property of the fractional maximal function of the measure.

The paper deals with the equation \(-\Delta u+a(x) u =|u|^{p-1}u \), \(u \in H^1({\mathbb {R}}^N)\), with \(N\ge 2\), \(p> 1,\ p< {N+2\over N-2}\) if \(N\ge 3\), \(a\in L^{N/2}_{loc}({\mathbb {R}}^N)\), \(\inf a> 0\), \(\lim _{|x| \rightarrow \infty } a(x)= a_\infty \). Assuming that the potential a(x) satisfies \(\lim _{|x| \rightarrow \infty }[a(x)-a_\infty ] e^{\eta |x|}= \infty \ \ \forall \eta

We prove that the support of an m dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered \( {\mathscr {H}}^{m} \) almost everywhere by a countable union of m dimensional submanifolds of class \( {\mathcal {C}}^{2} \). The \( {\mathcal {C}}^{2} \)-regularity of the submanifolds is optimal.

We consider autonomous integral functionals of the form $$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx \quad \text{ where } u:\varOmega \rightarrow {\mathbb {R}}^N, N\ge 1, \end{aligned}$$ where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of \({\mathcal {F}}\) assuming \(\fra

Proposition 3.1 of our manuscript (arXiv:1410.7303) cited Theorem 1.1 of Cao-Tran announced paper in 2013 (arXiv:1311.0846).

We prove a bubble tree convergence theorem for a sequence of closed Hamiltonian stationary Lagrangian surfaces with bounded areas and Willmore energies in a complete Kähler surface. We also prove two strong compactness theorems on the space of Hamiltonian stationary Lagrangian tori in \({\mathbb {C}}^2\) and \({{\mathbb {C}}}{{\mathbb {P}}}^2\) respectively.

In the present work, we established almost-sharp error estimates for linear elasticity systems in periodically perforated domains. The first result was \(L^{\frac{2d}{d-1-\tau }}\)-error estimates \(O\big (\varepsilon ^{1-\frac{\tau }{2}}\big )\) for all \(\tau \in (0,1)\) in a bounded smooth domain, which is new even for homogenization problems on unperforated domains. It followed from weighted Hardy–Sobolev’s

We consider the inhomogeneous Allen–Cahn equation $$\begin{aligned} \epsilon ^2\Delta u\,+\,V(y)(1-u^2)\,u\,=\,0\quad \text{ in }\ \Omega , \qquad \frac{\partial u}{\partial \nu }\,=\,0\quad \text{ on }\ \partial \Omega , \end{aligned}$$ where \(\Omega \) is a bounded domain in \({\mathbb {R}}^2\) with smooth boundary \(\partial \Omega \) and V(x) is a positive smooth function, \(\epsilon >0\) is a

We will be concerned with the existence of homoclinics for Lagrangian systems in \({\mathbb {R}}^N\) (\(N\ge 3 \)) of the form \(\frac{d}{dt}\left( \nabla \Phi (\dot{u}(t))\right) +\nabla _{u}V(t,u(t))=0\), where \(t\in {\mathbb {R}}\), \(\Phi {:}\,{\mathbb {R}}^N\rightarrow [0,\infty )\) is a G-function in the sense of Trudinger, \(V{:}\,{\mathbb {R}}\times \left( {\mathbb {R}}^N{\setminus }\{\xi

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding

We consider the nonlinear Stefan problem $$\begin{aligned} \left\{ \begin{array} {ll} u_t-d \Delta u=a u-b u^2 &{} \hbox {for}\quad x \in \Omega (t), \; t>0,\\ u=0\,\, \hbox {and}\,\,u_t=\mu |\nabla _x u |^2 &{}\hbox {for}\quad x \in \partial \Omega (t), \; t>0, \\ u(0,x)=u_0 (x) &{} \hbox {for}\quad x \in \Omega _0, \end{array} \right. \end{aligned}$$ where \(\Omega (0)=\Omega _0\) is an unbounded

In this work we introduce volume constraint problems involving the nonlocal operator \((-\Delta )_{\delta }^{s}\), closely related to the fractional Laplacian \((-\Delta )^{s}\), and depending upon a parameter \(\delta >0\) called horizon. We study the associated linear and spectral problems and the behavior of these volume constraint problems when \(\delta \rightarrow 0^+\) and \(\delta \rightarrow

We consider a one-dimensional discrete particle system of two species coupled through nonlocal interactions driven by the Newtonian potential, with repulsive self-interaction and attractive cross-interaction. After providing a suitable existence theory in a finite-dimensional framework, we explore the behaviour of the particle system in case of collisions and analyse the behaviour of the solutions

In this paper, we study the well-posedness of boundary layer problems for the inhomogeneous incompressible magnetohydrodynamics (MHD) equations, which are derived from the two-dimensional density-dependent incompressible MHD equations. Under the assumption that initial tangential magnetic field is not zero and density is a small perturbation of the outer constant flow in supernorm, the local-in-time

In the previous work Hamamoto (Calc Var Partial Differ Equ 58(4):23, 2019), following from an idea of Costin–Maz’ya (Costin and Maz’ya in Calc Var Partial Differ Equ 32(4):523–532, 2008), we computed the best constant in Rellich–Leray inequality for axisymmetric solenoidal fields, including any radial power weight. In the present paper, we recompute it without such a symmetry assumption. As a result

We investigate the next Trudinger–Moser critical equations, $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=\lambda ue^{u^2+\alpha |u|^\beta }&{}\text { in }B,\\ u=0&{}\text { on }\partial B, \end{array}\right. } \end{aligned}$$ where \(\alpha >0\), \((\lambda ,\beta )\in (0,\infty )\times (0,2)\) and \(B\subset {\mathbb {R}}^2\) is the unit ball centered at the origin. We classify the asymptotic

We prove global \(W^{1,q}({\varOmega },{\mathbb {R}}^m)\)-regularity for minimisers of convex functionals of the form \({\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x\).\(W^{1,q}({\varOmega },{\mathbb {R}}^m)\) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform \(\alpha \)-Hölder continuity assumption in x and controlled

We establish global well-posedness of strong solutions to the nonhomogeneous heat conducting magnetohydrodynamic equations with non-negative density on the whole space \({\mathbb {R}}^2\). More precisely, under compatibility conditions for the initial data, we show the global existence and uniqueness of strong solutions. Our method relies on delicate energy estimates and a logarithmic interpolation

We study the well-posedness of the vector-field Peierls–Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls–Nabarro model is reduced to a nonlocal scalar Ginzburg–Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson’s ratio

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in Band

Our aim is to characterize the homogeneous fractional Sobolev–Slobodeckiĭ spaces \(\mathcal {D}^{s,p} (\mathbb {R}^n)\) and their embeddings, for \(s \in (0,1]\) and \(p\ge 1\). They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo–Slobodeckiĭ seminorms. For \(s\,p < n\) or \(s = p = n = 1\) we show that \(\mathcal {D}^{s,p}(\mathbb

We study the Schrödinger–Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi u=a\left( x\right) \left| u\right| ^{p-2}u &{} \text { in }{{\mathbb {R}}}^{3}, \\ -\Delta \phi =u^{2} &{} \ \text {in }{{\mathbb {R}}}^{3}, \end{array} \right. \end{aligned}$$ where parameter \(\lambda >0\), \(20\) and other suitable conditions, we explore the energy functional corresponding

For a bounded open set \(\Omega \subset {\mathbb {R}}^3\) we consider the minimization problem $$\begin{aligned} S(a+\epsilon V) = \inf _{0\not \equiv u\in H^1_0(\Omega )} \frac{\int _\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int _\Omega u^6\,dx)^{1/3}} \end{aligned}$$ involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under

We define the elastic energy of smooth immersed closed curves in \({\mathbb {R}}^n\) as the sum of the length and the \(L^2\)-norm of the curvature, with respect to the length measure. We prove that the \(L^2\)-gradient flow of this energy smoothly converges asymptotically to a critical point. One of our aims was to the present the application of a Łojasiewicz–Simon inequality, which is at the core

In this paper, we prove that 2 dimensional transversal small perturbations of d-dimensional Euclidean planes under the skew mean curvature flow lead to global solutions which converge to the unperturbed planes in suitable norms. And we clarify the long time behaviors of the solutions in Sobolev spaces.

We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, \(\det D u =f\), where f is integrable and bounded away from zero. In particular, we take \(f\in L^p\), where \(p>1\), or in \(L\log L\). We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

The group \({\text {Diff}}({\mathcal {M}})\) of diffeomorphisms of a closed manifold \({\mathcal {M}}\) is naturally equipped with various right-invariant Sobolev norms \(W^{s,p}\). Recent work showed that for sufficiently weak norms, the geodesic distance collapses completely (namely, when \(sp\le \dim {\mathcal {M}}\) and \(s<1\)). But when there is no collapse, what kind of metric space is obtained

We study the saddle type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u = (I_\alpha *|u|^{\frac{N+\alpha }{N-2}} )|u|^{\frac{N+\alpha }{N-2}-2}u \quad \text { in }\;{\mathbb {R}}^N, \end{aligned}$$ where \(N\ge 3\), \(I_\alpha \) is the Riesz potential of order \(\alpha \in (0, N)\) and \(\frac{N+\alpha }{N-2}\) refers to the upper critical exponent with respect to the Hardy–Littlewood–Sobolev

We consider the nonlinear half-Laplacian heat equation $$\begin{aligned} u_t+(-\Delta )^{\frac{1}{2}} u-|u|^{p-1}u=0,\quad {\mathbb {R}}^n\times (0,T). \end{aligned}$$ We prove that all blows-up are type I, provided that \(n \le 4\) and \( 1

In this paper, we provide an extension to the whole euclidean space \( {\mathbb {R}}^N,\ N \ge 2, \) of the Trudinger–Moser inequalities proved by Calanchi and Ruf (Nonlinear Anal 121:403–411, 2015) involving a logarithmic weight. The inequalities are new and highlight very well the importance of the presence of this type of weight. Next, we prove some version of the concentration-compactness principle

We study asymptotic behavior of singular solutions of \(\Delta u+e^u=0\) in \(B \backslash \{0\}\), where \(B=\{x \in {\mathbb {R}}^2: \; |x|<1\}\). It is known that if \(u \in C^2(B \backslash \{0\})\) with \(\int _{B \backslash \{0\}} e^u dx<\infty \) is a singular solution to the equation, there is \(\alpha _0>-2\) such that $$\begin{aligned} u(x)=\alpha _0 \ln |x|+O(1) \;\; \text{ as } |x| \rightarrow

We prove a \(C^m\) Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is \(m-1\) times \(L^1\) differentiable almost everywhere coincides with a \(C^m\) horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if \(L^1\) differentiability is replaced