Parabolic Capacity and Soft Measures for Nonlinear Equations

We first introduce, using a functional approach, the notion of capacity related to the parabolic p-Laplace operator. Then we prove a decomposition theorem for measures (in space and time) that do not charge the sets of null capacity. We apply this result to prove existence and uniqueness of renormalized solutions for nonlinear parabolic initial boundary-value problems with such measures as right-hand side.     

Some noncoercive parabolic equations with lower order terms in divergence form

This paper deals with existence and regularity results for the problem $ \cases{u_t-\mathrm{div}(a(x,t,u )\nabla u)=-\mathrm{div}(u\,E) \qquad in \Omega\times (0,T),\cr u=0 \qquad on \partial \Omega\times (0,T), \cr u (0)= u_0 \qquad in \Omega ,\cr} $ under various assumptions on E and $ u_0 $. The main difculty in studying this problem is due to the presence of the term div(uE), which makes the differential operator non coercive on the "energy space" $ L^2 (0, T; H_0^1 (\Omega)) $.AMS Subject Classification: 35K10, 35K15, 35K65.     

Global Lipschitz regularizing effects for linear and nonlinear parabolic equations

In this paper we prove global bounds on the spatial gradient of viscosity solutions to second order linear and nonlinear parabolic equations in (0,T)×R^N$(0,T)\times {\mathbb{R}}^N$. Our assumptions include the case that the coefficients be both unbounded and with very mild local regularity (possibly weaker than the Dini continuity), the estimates only depending on the parabolicity constant and on the modulus of continuity of coefficients (but not on their L^∞$L^{\infty }$-norm). Our proof provides the analytic counterpart to the probabilistic proof used in Priola and Wang (2006) [35] (J. Funct. Anal. 2006) to get this type of gradient estimates in the linear case. We actually extend such estimates to the case of possibly unbounded data and solutions as well as to the case of nonlinear operators including Bellman–Isaacs equations. We investigate both the classical regularizing effect (at time t>0$t>0$) and the possible conservation of Lipschitz regularity from t=0$t=0$, and similarly we prove global Hölder estimates under weaker assumptions on the coefficients. The estimates we prove for unbounded data and solutions seem to be new even in the classical case of linear equations with bounded and Hölder continuous coefficients. Applications to Liouville type theorems are also given in the paper. Finally, we compare in an appendix the analytic and the probabilistic approach discussing the analogy between the doubling variables method of viscosity solutions and the probabilistic coupling method.     

Existence results for nonlinear parabolic equations via strong convergence of truncations

Summary  We prove existence results for the initial-boundary value problem for parabolic equations of the type

Elliptic equations with vertical asymptotes in the nonlinear term

We study the existence of solutions of the nonlinear problem {fx349-1} where μ is a bounded measure andg is a continuous nondecreasing function such thatg(0)=0. In this paper, we assume that the nonlinearityg satisfies {fx349-2} Problem (0.1) need not have a solution for every measure μ. We prove that, given μ, there exists a “closest” measure μ^* for which (0.1) can be solved. We also explain how assumption (0.2) makes problem (0.1) different from the case whereg(t) is defined for everyt ∈ ℝ.     

Gradient Bounds for Elliptic Problems Singular at the Boundary

Let Ω be a bounded smooth domain in \({{R}^N, N \geqq 2}\), and let us denote by d(x) the distance function d(x, ?Ω). We study a class of singular Hamilton–Jacobi equations, arising from stochastic control problems, whose simplest model is

$ - \alpha \Delta u+ u + \frac{\nabla u \cdot B (x)}{d (x)}+ c(x) |\nabla u|^2=f (x) \quad {\rm in}\,\Omega, $

where f belongs to \({W^{1,\infty}_{\rm loc} (\Omega)}\) and is (possibly) singular at \({\partial \Omega, c\in W^{1,\infty} (\Omega)}\) (with no sign condition) and the field \({B\in W^{1,\infty} (\Omega)^N}\) has an outward direction and satisfies \({B\cdot \nu\geqq \alpha}\) at ?Ω (ν is the outward normal). Despite the singularity in the equation, we prove gradient bounds up to the boundary and the existence of a (globally) Lipschitz solution. We show that in some cases this is the unique bounded solution. We also discuss the stability of such estimates with respect to α, as α vanishes, obtaining Lipschitz solutions for first order problems with similar features. The main tool is a refined weighted version of the classical Bernstein method to get gradient bounds; the key role is played here by the orthogonal transport component of the Hamiltonian.