This paper presents two observability inequalities for the heat equation over . In the first one, the observation is from a subset of positive measure in , while in the second, the observation is from a subset of positive surface measure on . It also proves the Lebeau-Robbiano spectral inequality when is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.
In this paper, we make some observations on the work of Di Fazio concerning estimates, , for solutions of elliptic equations , on a domain with Dirichlet data whenever and . We weaken the assumptions allowing real and complex non-symmetric operators and boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose...
We deal with a class of Penrose-Fife type phase field models for phase transitions, where the phase dynamics is ruled by a Cahn-Hilliard type equation. Suitable assumptions on the behaviour of the heat flux as the absolute temperature tends to zero and to are considered. An existence result is obtained by a double approximation procedure and compactness methods. Moreover, uniqueness and regularity results are proved as well.
2000 Mathematics Subject Classification: Primary 26A33; Secondary 47G20, 31B05We study a singular value problem and the boundary Harnack principle for the fractional Laplacian on the exterior of the unit ball.