Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation

A kind of supersolutions of the so-called p-parabolic equation are studied. These p-superparabolic functions are defined as lower semicontinuous functions obeying the comparison principle. Incidentally, they are precisely the viscosity supersolutions. One of our results guarantees the existence of a spatial Sobolev gradient. For p = 2 we have the supercaloric functions and the ordinary heat equation. Mathematics Subject Classification (2000) 35K55     

The Cartan,Choquet and Kellogg properties for the fine topology on metric spaces

We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.     

Supersolutions to Degenerate Elliptic Equations on Quasi open Sets

A nonliner potenitial theory for the supersolutions of equations similar to the ^p-Laplacian is studied on quasi open setes. We also introduce Sobolev spaces W₁, p(U) and W_{1 p} ^o on quasi sets. The equivalence of several possible definitions of such spaces, as well as completencess, density results and an analogue of Baby's theorem, are proved. The basic probperties of supersolutions to the above equation on quasi open sets and corresponding finely A-superharmonic functions (a nonlinear version of Fugleds's concept) are investigated. We prove that each bounded finely superharmonic function is a fine supersoluan asymptotic corrdinate path along which an unbounded A-subharmonie function grows to indinity     

On the Cauchy problem for systems —Parabolic systems—

The author proposes the definition of thep-parabolic system, which is stable under the similar transformations, using thep-determinant of the matrix of differential operators. The relation between thep-parabolic systems andH ^∞ well-posedness is considered. To the memory of Lamberto Cattabriga     

Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with { p^?,h^? } \left\{ {\overrightarrow p, \overrightarrow h } \right\} -parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.     

Reverse Khas’minskii condition

The aim of this paper is to present and discuss some equivalent characterizations of p-parabolicity for complete Riemannian manifolds in terms of existence of special exhaustion functions. In particular, Khas’minskii in Ergodic properties of recurrent diffusion prossesses and stabilization of solution to the Cauchy problem for parabolic equations (Theor Prob Appl 5(2), 1960) proved that if there exists a 2-superharmonic function ${\mathcal{K}}$ defined outside a compact set on a complete Riemannian manifold R such that ${\lim_{x\to \infty} \mathcal{K}(x)=\infty}$ , then R is 2-parabolic, and Sario and Nakai in Classification theory of Riemann surfaces (Springer, Berlin, 1970) were able to improve this result by showing that R is 2-parabolic if and only if there exists an Evans potential, i.e. a 2-harmonic function ${E:R{\setminus} K \to \mathbb{R}^+}$ with ${\lim_{x\to \infty}\mathcal{E}(x)=\infty}$ . In this paper, we will prove a reverse Khas’minskii condition valid for any p?>?1 and discuss the existence of Evans potentials in the nonlinear case.     

Cauchy problem for a class of degenerate kolmogorov-type parabolic equations with nonpositive genus

We study the properties of the fundamental solution and establish the correct solvability of the Cauchy problem for a class of degenerate Kolmogorov-type equations with { p^?,h^? } \left\{ {\overrightarrow p, \overrightarrow h } \right\} -parabolic part with respect to the main group of variables and nonpositive vector genus in the case where the solutions are infinitely differentiable functions and their initial values are generalized functions in the form of Gevrey ultradistributions.     

On the existence of solutions of the Dirichlet problem for nonlinear elliptic equations

In this paper we investigate the Dirichlet problem (1), (2) with the boundary data ?∈L ^∞(Q) and the nonlinearityb(x, u, p) having a quadratic growh inp. Using the method of sub- and supersolutions we prove the existence of a solution in a weighted Sobolev spaceW ^1,2(Q).     

Global Comparison Principles for the <Emphasis Type="Italic">p</Emphasis>-Laplace Operator on Riemannian Manifolds

We prove global comparison results for the p-Laplacian on a p-parabolic manifold. These involve both real-valued and vector-valued maps with finite p-energy. Further L ^q comparison principles in the non-parabolic setting are also discussed.     

Nonlinear superharmonic functions and supersolutions

Due to the lack of representation formulas for superharmonic functions associated with p-harmonic equations ${-\nabla \cdot(|\nabla u|^{p-2}\nabla u) = \mu}$ and their generalizations ${-\nabla \cdot A(x,\nabla u) = \mu}$ ,where ${A(x,\nabla u) \cdot \nabla u \approx | \nabla u |^{p}}$ , the interplay between nonlinear superharmonic functions and supersolutions is more important than in the linear case. Using the recent result of Kilpeläinen et. al., we establish sufficient and necessary conditions in terms of the Riesz measure μ that a p-superharmonic function is an ordinary weak supersolution. As an example we consider p-superharmonic solutions of the Poisson-type equation ${-\nabla \cdot A(x,\nabla u) = f(x)}$ .     

Positive solutions of the p-Laplacian involving a superlinear nonlinearity with zeros

Using a combination of several methods, such as variational methods, the sub and supersolutions method, comparison principles and a priori estimates, we study existence, multiplicity, and the behavior with respect to λ of positive solutions of p-Laplace equations of the form −Δpu=λh(x,u), where the nonlinear term has p-superlinear growth at infinity, is nonnegative, and satisfies h(x,a(x))=0 for a suitable positive function a. In order to manage the asymptotic behavior of the solutions we extend a result due to Redheffer and we establish a new Liouville-type theorem for the p-Laplacian operator, where the nonlinearity involved is superlinear, nonnegative, and has positive zeros.     

Liouville Type Theorems for Elliptic Equations with Gradient Terms

In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem ${-\Delta u + b(x)|\nabla u| = c(x)u}$ in exterior domains of ${\mathbb{R}^N}$ . We show that if lim ${{\rm inf}_{x \longrightarrow \infty} 4c(x) - b(x)^2 > 0}$ then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems ${-\Delta u + b|x|^{\lambda}|{\nabla} u| = c|x|^{\mu} u^p}$ and ${-\Delta u + be^{\lambda |x|}|\nabla u| = ce^{\mu |x|}u^p}$ , where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.     

On Backward p(x)-Parabolic Equations for Image Enhancement

In this study, we investigate the backward p(x)-parabolic equation as a new methodology to enhance images. We propose a novel iterative regularization procedure for the backward p(x)-parabolic equation based on the nonlinear Landweber method for inverse problems. The proposed scheme can also be extended to the family of iterative regularization methods involving the nonlinear Landweber method. We also investigate the connection between the variable exponent p(x) in the proposed energy functional and the diffusivity function in the corresponding Euler-Lagrange equation. It is well known that the forward problems converges to a constant solution destroying the image. The purpose of the approach of the backward problems is twofold. First, solving the backward problem by a sequence of forward problems, we obtain a smooth image which is denoised. Second, by choosing the initial data properly, we try to reduce the blurriness of the image. The numerical results for denoising appear to give improvement over standard methods as shown by preliminary results.     

Functional equations of spherical functions on p-adic homogeneous spaces

LetG be a connected reductive linear algebraic group andX aG-homogeneous affine algebraic variety both defined over a p-adic field k, where we assume a minimalk-parabolic subgroup ofG acts with open orbit. We are interested in spherical functions onX =X(k). In the present papaer, we give a unified method to obtain functional equations of spherical functions on X under the condition (AF) in the introduction, and explain functional equations are reduced to those ofp-adic local zeta functions of small prehomogeneous vector spaces of limited type.     

Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle

We introduce a new method for proving the nonexistence of positive supersolutions of elliptic inequalities in unbounded domains of ? ⁿ . The simplicity and robustness of our maximum principle-based argument provides for its applicability to many elliptic inequalities and systems, including quasilinear operators such as the p-Laplacian, and nondivergence form fully nonlinear operators such as Bellman-Isaacs operators. Our method gives new and optimal results in terms of the nonlinear functions appearing in the inequalities, and applies to inequalities holding in the whole space as well as exterior domains and cone-like domains.     

The porous medium equation with measure data

We study the existence of solutions to the porous medium equation with a nonnegative, finite Radon measure on the right-hand side. We show that such problems have solutions in a wide class of supersolutions. These supersolutions are defined as lower semicontinuous functions obeying a parabolic comparison principle with respect to continuous solutions. We also consider the question of how the integrability of the gradient of solutions is affected if the measure is given by a function in L ^s , for a small exponent s > 1.     

The Fundamental Convergence Theorem for <Emphasis Type="Italic">p</Emphasis>(·)-Superharmonic Functions

We study the balayage related to the supersolutions of the variable exponent p(·)-Laplace equation. We prove the fundamental convergence theorem for the balayage and apply it for proving the Kellogg property, boundary regularity results for the balayage, and a removability theorem for p(·)-solutions.     

Principle of localization of solutions of the Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type

In the case where initial data are generalized functions of the Gevrey-distribution type for which the classical notion of equality of two functions on a set is well defined, we establish the principle of local strengthening of the convergence of a solution of the Cauchy problem to its limit value as t → +0 for one class of degenerate parabolic equations of the Kolmogorov type with 2^?b \overrightarrow {2b} -parabolic part whose coefficients are continuous functions that depend only on t.     

Existence of strong solutions of Pucci extremal equations with superlinear growth in Du

We prove existence of strong solutions of Pucci extremal equations with superlinear growth in Du and unbounded coefficients. We apply this result to establish the weak Harnack inequality for L^p-viscosity supersolutions of fully nonlinear uniformly elliptic PDEs with superlinear growth terms with respect to Du.      

A strong maximum principle for weak solutions of quasi-linear elliptic equations with applications to Lorentzian and Riemannian geometry

The strong maximum principle is proved to hold for weak (in the sense of support functions) sub- and supersolutions to a class of quasi-linear elliptic equations that includes the mean curvature equation for C⁰-space-like hypersurfaces in a Lorentzian manifold. As one application, a Lorentzian warped product splitting theorem is given. © 1998 John Wiley & Sons, Inc.