The periodic parabolic eigenvalue problem with $L^\infty$ weight.

Some results on principal eigenvalues for periodic parabolic problems with weight

“…In [2], Beltramo and Hess proved, for C 2+s bounded domains, weights m € C e>e^2 (Q x R) and operators in nondivergence form, that if f T (1.2) P(m) := ess sup m(x,t)dt Jo *en then P(m) > 0 is a necessary and sufficient condition on m for the existence, uniqueness and (algebraic) simplicity of a positive principal eigenvalue. Later on there were many extensions of their results, weakening the regularity assumptions in the weight and the domain and also allowing more general boundary conditions (for example, [1,9,3,4,8] and the references therein). In all these works it is always assumed that the independent coefficient OQ is nonnegative.…”

“…And the alternative proof that it is known uses the weak form of the equation with a suitable test function, and so, or the coefficients a^ are assumed to be Cj. (Q x R) and then the operator can be rewritten in divergence form (for example, [9,7]), or the problem starts with an operator in divergence form (for example [3,4,8]). We suppose that this should be true for a general operator and m not continuous, but we do not know the proof.…”

Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources

“…where L is a linear uniformly elliptic operator, m(x, t) is a given weight function, see [5,9,10,13] and the original work of Beltramo and Hess [2,3]. We are quite interested in the case of weighted flux, in which (1) may have degeneracy or singularity due to the weight |x| α and the p-Laplacian when p = 2.…”

On Principal Eigenvalues for Periodic Parabolic Problems with Optimal Condition on the Weight Function

“…In [2], Beltramo extended these results to more general boundary conditions (that include the Neumann condition). The case m ∈ L ∞ T is solved in [10] with some additional regularity assumptions on the coefficients a i j , and an extension of these results for weights m ∈ L r T r > N + 2, can be found in [9]. On the other hand, for general bounded domains and operators with an elliptic part in divergence form, the case where m is an essentially bounded and lower semicontinuous function was studied by Daners in [5], and a latter extension for m ∈ L ∞ T appears in [7].…”