Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.     

Necessary and sufficient conditions for a curve to be the gradient range of a <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth function

We find some necessary and sufficient conditions for a plane curve to be the gradient range of a C ¹-smooth function of two variables. As one of the consequences we give the necessary and sufficient conditions on a continuous function ? under which the differential equation \(\frac{{\partial v}}{{\partial t}} = \varphi \left( {\frac{{\partial v}}{{\partial x}}} \right)\) has nontrivial C ¹-smooth solutions.     

Solutions of systems of elliptic differential equations on circular domains

We study global bifurcation of weak solutions of systems of elliptic differential equations considered on $\mathit{SO}(2)$-invariant domains. We formulate sufficient conditions for the existence of unbounded continua of nontrivial solutions branching from the trivial ones. As the main tool we use the degree for $\mathit{SO}(2)$-equivariant gradient maps defined by the second author in Rybicki (Nonlinear Anal. TMA 23(1) (1994) 83).     

Gradient estimates on the weighted p-Laplace heat equation

In this paper, by a regularization process we derive new gradient estimates for positive solutions to the weighted p-Laplace heat equation when the m-Bakry–Émery curvature is bounded from below by ?K for some constant $K\ge 0$. When the potential function is constant, which reduce to the gradient estimate established by Ni and Kotschwar for positive solutions to the p-Laplace heat equation on closed manifolds with nonnegative Ricci curvature if $K\searrow 0$, and reduce to the Davies, Hamilton and Li–Xu's gradient estimates for positive solutions to the heat equation on closed manifolds with Ricci curvature bounded from below if $p=2$.     

Gradient Estimates for the Positive Solutions of $$\mathfrak {L}u=0$$ on Self-Shrinkers

In this paper, we investigate the positive solutions of \(\mathfrak {L}u=0\) on a self-shrinker. First, we prove a global gradient estimate for the positive solutions, and obtain a strong Liouville theorem. Then by the generalized Laplacian comparison theorem for the distance function on a self-shrinker, we derive a local gradient estimate for the positive solutions. At last, we collect some applications of the local gradient estimate for the positive solutions on self-shrinkers.     

Well-posedness for dislocation based gradient visco-plasticity with isotropic hardening

In this work we establish the well-posedness for infinitesimal dislocation based gradient viscoplasticity with isotropic hardening for general subdifferential plastic flows. We assume an additive split of the displacement gradient into non-symmetric elastic distortion and non-symmetric plastic distortion. The thermodynamic potential is augmented with a term taking the dislocation density tensor $\mathrm{Curl}p$ into account. The constitutive equations in the models we study are assumed to be of self-controlling type. Based on the self-controlling property the existence of solutions of quasi-static initial–boundary value problems under consideration is shown using a time-discretization technique and a monotone operator method.     

Existence of radially symmetric solutions of the inhomogeneous <Emphasis Type="Italic">p</Emphasis>-Laplace equation

We consider the Dirichlet problem for the inhomogeneous p-Laplace equation with p nonlinear source. New sufficient conditions are established for the existence of weak bounded radially symmetric solutions as well as a priori estimates of solution and of the gradient of solution. We obtain an explicit formula that shows the dependence of the existence of these solutions on the dimension of the problem, the size of the domain, the exponent p, the nonlinear source, and the exterior mass forces.     

Density-Dependent Incompressible Fluids with Non-Newtonian Viscosity

We study the system of PDEs describing unsteady flows of incompressible fluids with variable density and non-constant viscosity. Indeed, one considers a stress tensor being a nonlinear function of the symmetric velocity gradient, verifying the properties of p-coercivity and (p–1)-growth, for a given parameter p > 1. The existence of Dirichlet weak solutions was obtained in [2], in the cases p 12/5 if d = 3 or p 2 if d = 2, d being the dimension of the domain. In this paper, with help of some new estimates (which lead to point-wise convergence of the velocity gradient), we obtain the existence of space-periodic weak solutions for all p 2. In addition, we obtain regularity properties of weak solutions whenever p 20/9 (if d = 3) or p 2 (if d = 2). Further, some extensions of these results to more general stress tensors or to Dirichlet boundary conditions (with a Newtonian tensor large enough) are obtained.     

Gradient estimates for ut=ΔF(u) on manifolds and some Liouville-type theorems

In this paper, we first prove a localized Hamilton-type gradient estimate for the positive solutions of Porous Media type equations:$u_t=\mathrm{\Delta }F(u),$ with $F^{\prime }(u)>0$, on a complete Riemannian manifold with Ricci curvature bounded from below. In the second part, we study Fast Diffusion Equation (FDE) and Porous Media Equation (PME):$u_t=\mathrm{\Delta }\left(u^p\right),p>0,$ and obtain localized Hamilton-type gradient estimates for FDE and PME in a larger range of p than that for Aronson–Bénilan estimate, Harnack inequalities and Cauchy problems in the literature. Applying the localized gradient estimates for FDE and PME, we prove some Liouville-type theorems for positive global solutions of FDE and PME on noncompact complete manifolds with nonnegative Ricci curvature, generalizing Yau?s celebrated Liouville theorem for positive harmonic functions.     

Partial regularity results for subelliptic systems in the Heisenberg group

We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial Hölder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the $\mathcal {A}We consider subelliptic systems in the Heisenberg group. We give a new proof for the smoothness of solutions of inhomogeneous systems with constant coefficients. With this result, we prove partial H?lder continuity of the horizontal gradient for non-linear systems with p-growth for p≥2 via the -harmonic approximation technique.     

Contraction of general transportation costs along solutions to Fokker–Planck equations with monotone drifts

We shall prove new contraction properties of general transportation costs along nonnegative measure-valued solutions to Fokker–Planck equations in ${\mathbb{R}}^d$, when the drift is a monotone (or λ-monotone) operator. A new duality approach to contraction estimates has been developed: it relies on the Kantorovich dual formulation of optimal transportation problems and on a variable-doubling technique. The latter is used to derive a new comparison property of solutions of the backward Kolmogorov (or dual) equation. The advantage of this technique is twofold: it directly applies to distributional solutions without requiring stronger regularity, and it extends the Wasserstein theory of Fokker–Planck equations with gradient drift terms, started by Jordan, Kinderlehrer and Otto (1998) [14], to more general costs and monotone drifts, without requiring the drift to be a gradient and without assuming any growth conditions.     

An efficient gradient method with approximate optimal stepsize for large-scale unconstrained optimization

In this paper, we introduce a new concept of approximate optimal stepsize for gradient method, use it to interpret the Barzilai-Borwein (BB) method, and present an efficient gradient method with approximate optimal stepsize for large unconstrained optimization. If the objective function f is not close to a quadratic on a line segment between the current iterate x _k and the latest iterate x _k?1, we construct a conic model to generate the approximate optimal stepsize for gradient method if the conic model is suitable to be used. Otherwise, we construct a new quadratic model or two other new approximation models to generate the approximate optimal stepsize for gradient method. We analyze the convergence of the proposed method under some suitable conditions. Numerical results show the proposed method is very promising.     

Existence of periodic solutions for a class of damped vibration problems

In this paper, we are concerned with the existence of periodic solutions for a class of damped vibration problems. By introducing some new kinds of superquadratic and asymptotically quadratic conditions, and making use of the generalized mountain pass theorem in critical point theory, we propose a unified approach when the potential function $F(t,x)$ exhibits either an asymptotically quadratic or a superquadratic behavior at infinity, and establish some sufficient conditions on periodic solutions, which extend and improve some recent results in the literature, even without damped vibration term.     

Nonexistence of positive weak solutions of elliptic inequalities

In this paper we give sufficient conditions for the nonexistence of positive entire weak solutions of coercive and anticoercive elliptic inequalities, both of the p$p$-Laplacian and of the mean curvature type, depending also on u$u$ and x$x$ inside the divergence term, while a gradient factor is included on the right-hand side. In particular, to prove our theorems we use a technique developed by Mitidieri and Pohozaev in [E. Mitidieri, S.I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001) 1–362], which relies on the method of test functions without using comparison and maximum principles. Their approach is essentially based first on a priori estimates and on the derivation of an asymptotics for the a priori estimate. Finally nonexistence of a solution is proved by contradiction.     

Yau’s gradient estimates for a nonlinear elliptic equation

Let \({(M^n,g)}\) be an n-dimensional complete Riemannian manifold. We consider Yau’s gradient estimates for positive solutions to the following nonlinear equation

$$\Delta u + au {\rm log} u=0$$

where a is a constant. As an application, we obtain the Liouville property for this equation in the case of a < 0. In addition, we illustrate, by giving concrete examples, that our results are sharp.

     

Porous medium equation and fast diffusion equation as gradient systems

We show that the Porous Medium Equation and the Fast Diffusion Equation, \(\dot u - \Delta {u^m} = f\), with m ∈ (0, ∞), can be modeled as a gradient system in the Hilbert space H ^?1(Ω), and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets Ω ? ? ⁿ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.     

A rigorous stability result for the Vlasov-Poisson system in three dimensions

It is proven that in a neutral two-component plasma with space homogeneous positively charged background, which is governed by the Vlasov-Poisson system and for which Poisson's equation is considered on a cube inR ³ with periodic boundary conditions, the space homogeneous stationary solutions g with energy gradient g/ 0 and compact support are (nonlinearly) stable in the L¹-norm with respect to weak solutions of the initial value problem.