Calculus of Variations and Partial Differential Equations最新文献

This work addresses several aspects of the dependence on p of the higher eigenvalues (lambda _n) to the Robin problem,

Here, (Omega subset {{mathbb {R}}}^N) is a (C^1) bounded domain, (nu ) is the outer unit normal, (Delta _p u = text {div} (|nabla u|^{p-2}nabla u)) stands for the p-Laplacian operator and (bin L^infty (partial Omega )). Main results concern: (a) the existence of the limits of (lambda _n) as (prightarrow 1), (b) the ‘limit problems’ satisfied by the ‘limit eigenpairs’, (c) the continuous dependence of (lambda _n) on p when (1< p <infty ) and (d) the limit profile of the eigenfunctions as (prightarrow 1). The latter study is performed in the one dimensional and radially symmetric cases. Corresponding properties on the Dirichlet and Neumann eigenvalues are also studied in these two special scenarios.

We revisit Yudovich’s well-posedness result for the 2-dimensional Euler equations for an inviscid incompressible fluid on either a sufficiently regular (not necessarily bounded) open set (Omega subset mathbb {R}^2) or on the torus (Omega =mathbb {T}^2). We construct global-in-time weak solutions with vorticity in (L^1cap L^p_{ul}) and in (L^1cap Y^Theta _{ul}), where (L^p_{ul}) and (Y^Theta _{ul}) are suitable uniformly-localized versions of the Lebesgue space (L^p) and of the Yudovich space (Y^Theta ) respectively, with no condition at infinity for the growth function (Theta ). We also provide an explicit modulus of continuity for the velocity depending on the growth function (Theta ). We prove uniqueness of weak solutions in (L^1cap Y^Theta _{ul}) under the assumption that (Theta ) grows moderately at infinity. In contrast to Yudovich’s energy method, we employ a Lagrangian strategy to show uniqueness. Our entire argument relies on elementary real-variable techniques, with no use of either Sobolev spaces, Calderón–Zygmund theory or Littlewood–Paley decomposition, and actually applies not only to the Biot–Savart law, but also to more general operators whose kernels obey some natural structural assumptions.

In this manuscript, given a metric tensor on the probability simplex, we define differential operators on the Wasserstein space of probability measures on a graph. This allows us to propose a notion of graph individual noise operator and investigate Hamilton–Jacobi equations on this Wasserstein space. We prove comparison principles for viscosity solutions of such Hamilton–Jacobi equations and show existence of viscosity solutions by Perron’s method. We also discuss a model optimal control problem and show that the value function is the unique viscosity solution of the associated Hamilton–Jacobi–Bellman equation.

Suppose (Omega _0,Omega _1) are two bounded strongly (mathbb {C})-convex domains in (mathbb {C}^n), with (nge 2) and (Omega _1supset overline{Omega _0}). Let (mathcal {R}=Omega _1backslash overline{Omega _0}). We call (mathcal {R}) a (mathbb {C})-convex ring. We will show that for a solution (Phi ) to the homogenous complex Monge–Ampère equation in (mathcal {R}), with (Phi =1) on (partial Omega _1) and (Phi =0) on (partial Omega _0), (sqrt{-1}partial {overline{partial }}Phi ) has rank (n-1) and the level sets of (Phi ) are strongly (mathbb {C})-convex.

We revisit the following nonlinear Schrödinger equation

where (varepsilon >0) is a small parameter, (Nge 2) and (1<p<2^*-1). It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive k-peak solutions to above problem when the critical points of V(x) are non-isolated and degenerate. We also give a specific formula for the Morse index of k-peak solutions when the critical point set of V(x) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential V(x). Our results generalize Grossi and Servadei’s work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions.

We construct optimal Hardy weights to subcritical energy functionals h associated with quasilinear Schrödinger operators on infinite graphs. Here, optimality means that the weight w is the largest possible with respect to a partial ordering, and that the corresponding shifted energy functional (h-w) is null-critical. Moreover, we show a decay condition of Hardy weights in terms of their integrability with respect to certain integral weights. As an application of the decay condition, we show that null-criticality implies optimality near infinity. We also briefly discuss an uncertainty-type principle, a Rellich-type inequality and examples.

In this paper, we classify the hypersurfaces of (mathbb {S}^2times mathbb {S}^2) with constant sectional curvature. We prove that the constant sectional curvature can only be (frac{1}{2}). We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in (mathbb {S}^2times mathbb {S}^2), and we establish a one-to-one correspondence between such minimal hypersurface and the solution to the famous “sinh-Gordon equation” ( left( frac{partial ^2}{partial u^2}+frac{partial ^2}{partial v^2}right) h =-tfrac{1}{sqrt{2}}sinh left( sqrt{2}hright) ).

We obtain a complete description of divergent Palais–Smale sequences for the prescribed Ricci curvature functional on compact homogeneous spaces. As an application, we prove the existence of saddle points on generalized Wallach spaces and several types of generalized flag manifolds. We also describe the image of the Ricci map in some of our examples.

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is (uequiv 0). We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions.

In this paper we consider the following focusing mass-critical Hartree equation with a defocusing perturbation and harmonic potential

where (Nge 3), (2<p<2^*={2N}/({N-2})) and (varepsilon >0). We mainly focus on the normalized ground state solitary waves of the form (psi (t,x)=e^{imu t}u_{varepsilon ,rho }(x)), where (u_{varepsilon ,rho }(x)) is radially symmetric-decreasing and (int _{mathbb {R}^N}|u_{varepsilon ,rho }|^2,dx=rho ). Firstly, we prove the existence and nonexistence of normalized ground states under the (L^2)-subcritical, (L^2)-critical ((p=4/N +2)) and (L^2)-supercritical perturbations. Secondly, we characterize perturbation limit behaviors of ground states (u_{varepsilon ,rho }) as (varepsilon rightarrow 0^+) and find that the (varepsilon )-blow-up phenomenon happens for (rho ge rho _c=Vert QVert ^2_{L^2}), where Q is a positive radially symmetric ground state of (-Delta u+u-(|x|^{-2}*|u|^2)u=0) in (mathbb {R}^N). We prove that (int _{mathbb {R}^N}|nabla u_{varepsilon ,rho }(x)|^2,dxsim varepsilon ^{-frac{4}{N(p-2)+4}}) for (rho =rho _c) and (2<p<2^*), while (int _{mathbb {R}^N}|nabla u_{varepsilon ,rho }|^2,dxsim varepsilon ^{-frac{4}{N(p-2)-4}}) for (rho >rho _c) and (4/N+2<p<2^*), and obtain two different blow-up profiles corresponding to two limit equations. Finally, we study the limit behaviors as (varepsilon rightarrow +infty ), which corresponds to a Thomas–Fermi limit. The limit profile is given by the Thomas–Fermi minimizer (u^{TF}=left[ mu ^{TF}-|x|^2 right] ^{frac{1}{p-2}}_{+}), where (mu ^{TF}) is a suitable Lagrange multiplier with exact value. Moreover, we obtain a sharp vanishing rate for (u_{varepsilon , rho }) that (Vert u_{varepsilon , rho }Vert _{L^{infty }}sim varepsilon ^{-frac{N}{N(p-2)+4}}) as (varepsilon rightarrow +infty ).