2,572 research outputs found
The classical overdetermined Serrin problem
In this survey we consider the classical overdetermined problem which was
studied by Serrin in 1971. The original proof relies on Alexandrov's moving
plane method, maximum principles, and a refinement of Hopf's boundary point
Lemma. Since then other approaches to the same problem have been devised. Among
them we consider the one due to Weinberger which strikes for the elementary
arguments used and became very popular. Then we discuss also a duality approach
involving harmonic functions, a shape derivative approach and a purely integral
approach, all of them not relying on maximum principle. For each one we
consider pros and cons as well as some generalizations
On a conjectured reverse Faber-Krahn inequality for a Steklov-type Laplacian eigenvalue
For a given bounded Lipschitz set , we consider a Steklov--type
eigenvalue problem for the Laplacian operator whose solutions provide extremal
functions for the compact embedding . We prove that a conjectured reverse Faber--Krahn inequality holds
true at least in the class of Lipschitz sets which are "close" to a ball in a
Hausdorff metric sense. The result implies that among sets of prescribed
measure, balls are local minimizers of the embedding constant
Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states
In this paper we study the global approximate multiplicative controllability
for nonlinear degenerate parabolic Cauchy problems. In particular, we consider
a one-dimensional semilinear degenerate reaction-diffusion equation in
divergence form governed via the coefficient of the \-reaction term (bilinear
or multiplicative control). The above one-dimensional equation is degenerate
since the diffusion coefficient is positive on the interior of the spatial
domain and vanishes at the boundary points. Furthermore, two different kinds of
degenerate diffusion coefficient are distinguished and studied in this paper:
the weakly degenerate case, that is, if the reciprocal of the diffusion
coefficient is summable, and the strongly degenerate case, that is, if that
reciprocal isn't summable. In our main result we show that the above systems
can be steered from an initial continuous state that admits a finite number of
points of sign change to a target state with the same number of changes of sign
in the same order. Our method uses a recent technique introduced for uniformly
parabolic equations employing the shifting of the points of sign change by
making use of a finite sequence of initial-value pure diffusion pro\-blems. Our
interest in degenerate reaction-diffusion equations is motivated by the study
of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers
model) and some models in population genetics (see, e.g., the Fleming-Viot
model).Comment: arXiv admin note: text overlap with arXiv:1510.0420
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