Nonlinear Analysis-Theory Methods & Applications最新文献

We establish the absence of the Lavrentiev gap between Sobolev and smooth maps for a non-autonomous variational problem of a general structure, where the integrand is assumed to be controlled by a function which is convex and anisotropic with respect to the last variable. This fact follows from new results on fine approximation properties of the natural underlying unconventional function space. Scalar and vector-valued problems are studied.

In this manuscript, we derive some Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations of the form: ${\partial }_{t}u-F(x,t,D^{2}u)=f(x,t)\text{in}{Q}_1=B_1\times (-1,0],$provided that the source $f$ and the coefficients of $F$ are Hólder continuous functions, and $F$ enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are $C^{1,\text{Log-Lip}}$-regular. We also obtain Calderón-Zygmund estimates for such a class of non-convex operators. Finally, we connect our results and recent estimates for fully nonlinear models in certain solution classes.

In this paper, we study the extended Ricci flow on a complete noncompact Riemannian manifold of dimension $n$ introduced by List in List (2008), and prove the short-time existence with bounded $L^p$ norm of Riemann curvature. In the critical case $p=\frac{n}{2}$, we replace the bounded $L^p$ norm of Riemann curvature by the bounded $L^p$ norm of Ricci curvature in the short-time existence.

In the context of Finsler manifolds, the paper explores the existence, asymptotic boundary behavior, and uniqueness of viscosity solutions to infinite boundary-value problems associated with the normalized infinite Laplacian in relatively compact subsets. The equation under consideration incorporates lower-order terms featuring non-linear gradient terms. To achieve this objective, we study Dirichlet problems with continuous boundary data and establish a comparison principle, which is of independent significance.

In this paper we demonstrate a sufficient condition for blowup of the nonlinear Klein–Gordon equation with arbitrarily positive initial energy in Friedmann–Lemaître–Robertson–Walker spacetimes. This is accomplished using an established concavity method that has been employed for similar PDEs in Minkowski space. This proof relies on the energy inequality associated with this equation, $E\left(t_0\right)\ge E\left(t\right)$, also proved herein using a geometric method.

In present paper, we study the following nonlinear Schrödinger equation with combined power nonlinearities $-\Delta u+V\left(x\right)u+\lambda u={\left|u\right|}^{2^{\ast }-2}u+\mu {\left|u\right|}^{q-2}u\text{in}{R}^N,N\ge 3$having prescribed mass ${\int }_{R^N}{u}^{2}dx=a^2,$where $\mu ,a>0$, $q\in (2,2^{\ast })$, $2^{\ast }=\frac{2N}{N-2}$ is the critical Sobolev exponent, $V$ is an external potential vanishing at infinity, and the parameter $\lambda \in R$ appears as a Lagrange multiplier. Under some mild assumptions on $V$, combining the Pohožaev manifold, constrained minimization arguments and some analytical skills, we get the existence of normalized solutions for the problem with $q\in (2,2^{\ast })$. At the same time, the exponential decay property of the solutions is established, which is important for the instability analysis of the standing waves. Furthermore, we give a description of the ground state set and obtain the strong instability of the standing waves for $q\in [2+\frac{4}{N},2^{\ast })$.

We prove existence for a class of signed Radon measure-valued entropy solutions of the Cauchy problem for a first order scalar hyperbolic conservation law in one space dimension. The initial data of the problem is a finite superposition of Dirac masses, whereas the flux is Lipschitz continuous. Existence is proven by a constructive procedure which makes use of a suitable family of approximating problems. Relevant qualitative properties of such constructed solutions are pointed out.

The aim of this paper is to investigate the minimization problem related to a Ginzburg–Landau energy functional, where in particular a nonlinear diffusion of mean curvature-type is considered, together with a classical double well potential. A careful analysis of the corresponding Euler–Lagrange equation, equipped with natural boundary conditions and mass constraint, leads to the existence of an unique Maxwell solution, namely a monotone increasing solution obtained for small diffusion and close to the so-called Maxwell point. Then, it is shown that this particular solution (and its reversal) has least energy among all the stationary points satisfying the given mass constraint. Moreover, as the viscosity parameter tends to zero, it converges to the increasing (decreasing for the reversal) single interface solution, namely the constrained minimizer of the corresponding energy without diffusion. Connections with Cahn–Hilliard models, obtained in terms of variational derivatives of the total free energy considered here, are also presented.

We consider a hydrodynamic model of flocking-type with all-to-all interaction kernel in one-space dimension and establish that the global entropy weak solutions, constructed in Amadori and Christoforou (2022) to the Cauchy problem for any $BV$ initial data that has finite total mass confined in a bounded interval and initial density uniformly positive therein, admit unconditional time-asymptotic flocking without any further assumptions on the initial data. In addition, we show that the convergence to a flocking profile occurs exponentially fast.

We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge–Ampère Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate which is from the Monge–Ampère eigenvalue problem. As a result, we obtain the sharp boundary estimates and the optimal global Hölder regularity for the two equations.