1,721,015 research outputs found
Steiner symmetrization: a weighted version of Polya - Szego principle
No full textLet us consider a Dirichlet integral of the type
F(u) =
Z
f(Du) dx
where u is a nonnegative Sobolev function with compact support in
. The well known P¶olya-SzegÄo principle states that if us denotes the
Steiner symmetrand of u, then:
F(us) · F(u):
We study the case when the integrand f depends also on x and u
Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states
No full textIn 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 problems. 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)
A quantitative version of the isoperimetric inequality: the anisotropic case
No full textWe state and prove a stability result for the anisotropic version of
the isoperimetric inequality. Namely if E is a set with small anisotropic
isoperimetric deficit, then E is ''close" to the Wulff shape set
Ceramica attica nei santuari della costa ionica dell’Italia meridionale: colonie achee e indigeni tra paralia e mesogaia
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