If is a nonnegative local weak solution of (1.2)–(1.4), then is locally bounded in Q.
If is a nonnegative local weak solution of (3.37), (1.3), and (1.4), then is locally bounded in Q.
In the present paper, we apply superdiffusions to a more general equationu+Lu=ψ u), whereLu=L0u+cu(with a bounded coefficientc) andψbelongs to a convex class which containskuαwith 1<α⩽2 and positive locally bounded coefficientk.
Also, we obtain Gaussian bound for Robin Green's function under an additional assumption that weak solutions of Robin problem are locally bounded up to the boundary.
This easily implies, for example, Cartan's classical theorem on the supremum of a locally bounded family of subharmonic functions.
We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics.
In this paper we prove the existence of locally bounded variation solution of a Volterra type set-valued integral equation of arbitrary (not necessarily integer) order.
We prove that the initial trace is an outer regular Borel measure which may not be locally bounded for some values of the parameters p and q.
For locally bounded semigroups T = {T t)} t ≥ 0, we obtain similar results in terms of the maps t ↦ ||T (t) x||.
We prove that minimizers of such non-differentiable functionals are locally bounded, Hölder continuous, and that they satisfy a suitable Harnack inequality.
Ifϕis a Borel function locally bounded above onΩ, and ifΩhas a Green's function, then for each x∈Ω,sup{v(x): vsubharmonic on Ω v⩽ϕ}=inf .We also investigate to what extent these results carry over to plurisubharmonic functions.
