726 research outputs found
Unique Continuation for Sublinear Elliptic Equations Based on Carleman Estimates
In this article we deal with different forms of the unique continuation
property for second order elliptic equations with nonlinear potentials of
sublinear growth. Under suitable regularity assumptions, we prove the weak and
the strong unique continuation property. Moreover, we also discuss the unique
continuation property from measurable sets, which shows that nodal domains to
these equations must have vanishing Lebesgue measure. Our methods rely on
suitable Carleman estimates, for which we include the sublinear potential into
the main part of the operator.Comment: 22 page
Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation
In this paper we establish some new Carleman estimates for the
Baouendi-Grushin operators , in (1.1) below. We apply such
estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique
continuation; (ii) the strong unique continuation property for some degenerate
sublinear equations.Comment: revised version of the file, several references have been adde
Existence, unique continuation and symmetry of least energy nodal solutions to sublinear Neumann problems
We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c
l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 &
\textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where is a bounded domain, and . For ,
will be identified with \sgn(u). We establish a variational
principle for least energy nodal solutions, and we investigate their
qualitative properties. In particular, we show that they satisfy a unique
continuation property (their zero set is Lebesgue-negligible). Moreover, if
is radial, then least energy nodal solutions are foliated Schwarz
symmetric, and they are nonradial in case is a ball. The case
requires special treatment since the formally associated energy functional is
not differentiable, and many arguments have to be adjusted
Beyond the classical strong maximum principle: forcing changing sign near the boundary and flat solutions
We show that the classical strong maximum principle, concerning positive
supersolutions of linear elliptic equations vanishing on the boundary of the
domain can be extended, under suitable conditions, to the case in
which the forcing term is changing sign. In addition, in the case of
solutions, the normal derivative on the boundary may also vanish on the
boundary (definition of flat solutions). This leads to examples in which the
unique continuation property fails. As a first application, we show the
existence of positive solutions for a sublinear semilinear elliptic problem of
indefinite sign. A second application, concerning the positivity of solutions
of the linear heat equation, for some large values of time, with forcing and/or
initial datum changing sign is also given.Comment: 20 pages 2 Figure
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