Calculus of Variations and Partial Differential Equations最新文献

For fixed positive integer n, (pin [0,1)), (ain (0,1)), we prove that if a function (g:{mathbb {S}}^{n-1}rightarrow {mathbb {R}}) is sufficiently close to 1, in the (C^a) sense, then there exists a unique convex body K whose (L_p) curvature function equals g. This was previously established for (n=3), (p=0) by Chen et al. (Adv Math 411(A):108782, 2022) and in the symmetric case by Chen et al. (Adv Math 368:107166, 2020). Related, we show that if (p=0) and (n=4) or (nle 3) and (pin [0,1)), and the (L_p) curvature function g of a (sufficiently regular, containing the origin) convex body K satisfies (lambda ^{-1}le gle lambda ), for some (lambda >1), then (max _{xin {mathbb {S}}^{n-1}}h_K(x)le C(p,lambda )), for some constant (C(p,lambda )>0) that depends only on p and (lambda ). This also extends a result from Chen et al. [10]. Along the way, we obtain a result, that might be of independent interest, concerning the question of when the support of the (L_p) surface area measure is lower dimensional. Finally, we establish a strong non-uniqueness result for the (L_p)-Minkowksi problem, for (-n<p<0).

We obtain existence results for the Q-curvature equation of order 2k on a closed Riemannian manifold of dimension (nge 2k+1), where (kge 1) is an integer. We obtain these results under the assumptions that the Yamabe invariant of order 2k is positive and the Green’s function of the corresponding operator is positive, which are satisfied in particular when the manifold is Einstein with positive scalar curvature. In the case where (2k+1le nle 2k+3) or the manifold is locally conformally flat, we assume moreover that the operator has positive mass. In the case where (nge 2k+4) and the manifold is not locally conformally flat, the results essentially reduce to the determination of the sign of a complicated constant depending only on n and k.

In this paper, we study the uniqueness of weak solutions of the heat flow of half-harmonic maps, which was first introduced by Wettstein as a half-Laplacian heat flow and recently studied by Struwe using more classical techniques. On top of its similarity with the two dimensional harmonic map flow, this geometric gradient flow is of interest due to its links with free boundary minimal surfaces and the Plateau problem, leading Struwe to propose the name Plateau flow, which we adopt throughout. We obtain uniqueness of weak solutions of this flow under a natural condition on the energy, which answers positively a question raised by Struwe.

We here investigate a modification of the compressible barotropic Euler system with friction, involving a fuzzy nonlocal pressure term in place of the conventional one. This nonlocal term is parameterized by (varepsilon > 0) and formally tends to the classical pressure when (varepsilon ) approaches zero. The central challenge is to establish that this system is a reliable approximation of the classical compressible Euler system. We establish the global existence and uniqueness of regular solutions in the neighborhood of the static state with density 1 and null velocity. Our results are demonstrated independently of the parameter (varepsilon ,) which enable us to prove the convergence of solutions to those of the classical Euler system. Another consequence is the rigorous justification of the convergence of the mass equation to various versions of the porous media equation in the asymptotic limit where the friction tends to infinity. Note that our results are demonstrated in the whole space, which necessitates to use the (L^1(mathbb {R}_+; dot{B}^sigma _{2,1}(mathbb {R}^d))) spaces framework.

We investigate regularity estimates of quasi-minima of the Alt–Caffarelli energy functional. We prove universal Hölder continuity of quasi-minima and optimal Lipchitz regularity along their free boundaries.

We consider an elliptic system of Hamiltonian type in a strip in ({mathbb {R}}^N), satisfying the periodic boundary condition for the first k variables. In the superlinear case with critical growth, we prove the existence of a single bubbling solution for the system under an optimal condition on k. The novelty of the paper is that all the estimates needed in the proof of the existence result can be obtained once the Green’s function of the Laplacian operator in a strip with periodic boundary conditions is found.

In this paper we consider an obstacle problem for a generalization of the p-elastic energy among graphical curves with fixed ends. Taking into account that the Euler–Lagrange equation has a degeneracy, we address the question whether solutions have a flat part, i.e. an open interval where the curvature vanishes. We also investigate which is the main cause of the loss of regularity, the obstacle or the degeneracy. Moreover, we give several conditions on the obstacle that assure existence and nonexistence of solutions. The analysis can be refined in the special case of the p-elastica functional, where we obtain sharp existence results and uniqueness for symmetric minimizers.

In this paper we provide a positive lower bound for the number of metrics of constant Q-curvature which are conformal to a Riemannian product of the form ((Mtimes X, g+delta h)), where (delta >0) is a small positive constant, (M, g) is a closed (compact without boundary) n-dimensional Riemannian manifold and (X, h) a closed m-dimensional (positive) Einstein manifold. We assume that (mge 3) and (nge 2) or, if (m=2), that (nge 7). More specifically, we study the constant Q-curvature equation on the Riemannian product ((Mtimes X, g+delta h)), which becomes, by restricting the equation to functions which depend only on the M-variable, a subcritical equation on (M, g) driven by a fourth order operator, known as the Paneitz operator. Then we prove that, for (delta >0) small enough, the equation has at least (textrm{Cat}(M)) positive solutions, where (textrm{Cat}(M)) is the Lusternik-Schnirelmann category of M. This implies that there are at least (textrm{Cat}(M)) metrics of constant Q-curvature in the conformal class of the Riemannian product ((Mtimes X, g+delta h)).

It is well known that isoperimetric regions in a smooth compact ((n+1))-manifold are themselves smooth, up to a closed set of codimension at most 8. In this note, we construct an 8-dimensional compact smooth manifold whose unique isoperimetric region with half volume that of the manifold exhibits two isolated singularities. This stands in contrast with the situation in which a manifold is a space form, where isoperimetric regions are smooth in every dimension.

In the previous paper (Ding et al. in J Differ Equ 385:183–253, 2024), for the 3D quasilinear wave equation (-big (1+(partial _tphi )^pbig )partial _t^2phi +Delta phi =0) with short pulse initial data ((phi ,partial _tphi )(1,x)=big (delta ^{2-varepsilon _{0}}phi _0 (frac{r-1}{delta },omega ),delta ^{1-varepsilon _{0}}phi _1(frac{r-1}{delta },omega )big )), where (pin mathbb {N}), (0<varepsilon _{0}<1), under the outgoing constraint condition ((partial _t+partial _r)^kphi (1,x)=O(delta ^{2-varepsilon _{0}-kmax {0,1-(1-varepsilon _{0})p}})) for (k=1,2), the authors establish the global existence of smooth large solution (phi ) when (p>p_c) with (p_c=frac{1}{1-varepsilon _{0}}). In the present paper, under the same outgoing constraint condition, when (1le ple p_c), we will show that the smooth solution (phi ) may blow up and further the outgoing shock is formed in finite time.