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164
We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary, requiring a (possibly nonlinear) bound on the normal derivative at the boundary. These inequalities give rise to an energy quantization principle for sequences of solutions of boundary
value problems that have bounded energy and whose energy densities satisfy nonlinear bounds on the Laplacian and normal...
We consider a class of semilinear elliptic equations of the formwhere , is a periodic, positive function and is modeled on the classical two well Ginzburg-Landau potential . We look for solutions to (1) which verify the asymptotic conditions as uniformly with respect to . We show via variational methods that if is sufficiently small and is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.
We consider a class of
semilinear elliptic equations of the form
15.7cm
-
where , is a periodic, positive function and
is modeled on the classical two well Ginzburg-Landau
potential . We look for solutions to ([see full textsee full text])
which verify the
asymptotic conditions as
uniformly with respect to .
We show via variational
methods that if ε is sufficiently small and a is not constant,
then ([see full textsee full text])
admits infinitely many of such solutions, distinct...
We study nonlinear elliptic equations of the form where the main assumption on and is that there exists a one dimensional solution which solves the equation in all the directions . We show that entire monotone solutions are one dimensional if their level set is assumed to be Lipschitz, flat or bounded from one side by a hyperplane.
We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain with a nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant, accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part . The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization...
We consider a nonlinear second order elliptic boundary
value problem (BVP)
in a bounded domain with
a nonlocal boundary condition.
A Dirichlet BC containing an unknown additive constant,
accompanied with a nonlocal (integral) Neumann side condition is
prescribed at some boundary part Γn.
The rest of the boundary is equipped with Dirichlet or nonlinear Robin
type BC. The solution is found via linearization. We design a robust and
efficient approximation scheme.
Error estimates for...
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