Weak Supersolutions Research Articles

Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations

In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.

Unbounded Supersolutions of Nonlinear Equations with Nonstandard Growth

We show that every weak supersolution of a variable exponent -Laplace equation is lower semicontinuous and that the singular set of such a function is of zero capacity if the exponent is logarithmically Holder continuous. As a technical tool we derive Harnack-type estimates for possibly unbounded supersolutions.

On a Liouville phenomenon for entire weak supersolutions of elliptic partial differential equations

We study a new Liouville-type phenomenon for entire weak supersolutions of elliptic partial differential equations of the form A(u)=0 on Rn, n≥2. Typical examples of the operator A(u) are the p-Laplacian for p>1, the mean curvature operator, and their well-known modifications.

On the positivity of weak supersolutions of non-uniformly elliptic equations

The classical maximum principle for homogeneous, second order, uniformly elliptic equations implies that non-negative, classical supersolutions are either positive or vanish identically in the interior of their domain of definition. This paper is concerned with an extension of this result to weak supersolutions of non-uniformly elliptic equations subject to only mild coefficient restrictions.

On semilinear elliptic boundary value problems in unbounded domains

SynopsisWe consider the possibility of solving semilinear elliptic boundary value problems in unbounded domains. We first treat the case when the non-linear terms are independent of terms involving gradients. Using a monotone iteration scheme, we show that the existence of a weak subsolution v and a weak supersolution w ≧ v, implies the existence of a weak solution u, and v ≦ u ≦ w. We also state conditions which guarantee the existence of a solution when only a subsolution is known to exist. Next, we suppose the non-linear terms can depend on gradient terms. Using a method developed in [4], based on perturbation theory of maximal monotone operators, we prove the existence of a H2(Ω) solution lying between a given H2(Ω) subsolution v and a given H2(Ω) supersolution w ≧ v.