scholarly journals Degenerate scale for 2D Laplace equation with mixed boundary condition and comparison with other conditions on the boundary

2018 ◽  
Vol 88 ◽  
pp. 14-25 ◽  
Author(s):  
A. Corfdir ◽  
G. Bonnet
Keyword(s):  
Boundary Condition ◽  
Laplace Equation ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Degenerate Scale

scholarly journals A strong maximum principle for the fractional laplace equation with mixed boundary condition

2021 ◽  
Vol 24 (6) ◽  
pp. 1699-1715
Author(s):  
Rafael López-Soriano ◽  
Alejandro Ortega
Keyword(s):  
Boundary Condition ◽  
Maximum Principle ◽  
Laplace Equation ◽  
Boundary Data ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Elliptic Problems ◽  
Neumann Boundary ◽  
Strong Maximum Principle ◽  
The One

Abstract In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet–Neumann boundary data which extends the one proved by J. Dávila (cf. [11]) to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non–local counterpart to a Hopf’s Lemma for fractional elliptic problems with mixed boundary data.

scholarly journals Boundary value controllability and observability problems for the wave and heat equation

1988 ◽  
Vol 29 (4) ◽  
pp. 461-479
Author(s):  
K.-D. Werner
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Mixed Boundary Condition ◽  
Mixed Boundary ◽  
Finite Energy ◽  
Duality Principle ◽  
Spherical Region ◽  
Initial States ◽  
The Fourier Transform ◽  
Controllability And Observability

AbstractIn this paper, we study controllability and observability problems for the wave and heat equation in a spherical region in Rn, where the control enters in the mixed boundary condition. In the main result, we show that all “finite energy” initial states (i.e. (ω0, ν0) ∈ H1(Ω) × L2 (Ω)) can be steered to zero at time T, using a control f ∈ L2 (∂Ω × [0, T]), provided T > 2. On this basis, we use the duality principle to investigate initial observability for the wave equation. Applying the Fourier transform technique, we obtain controllability and observability results for the heat equation.

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