Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Bolzano mean-value theorem and partial differential equations   We study the existence of solutions to abstract equations of the form $0 = Au
+ F(u)$, $u\in K\subset E$, where A is an abstract differential operator acting
in a Banach space $E$, $K$ is a closed convex set of constraints being
invariant with respect to resolvents of A and perturbations are subject to
different tangency condition. Such problems are closely related to the
so-called Poicar\'e- Miranda theorem, being the multi-dimensional counterpart
of the celebrated Bolzano intermediate value theorem. In fact our main results
can and should be regarded as infinite-dimensional variants of Bolzano and
Miranda-Poincar\'e theorems. Along with single-valued problems we deal with
set-valued ones, yielding the existence of the so-called constrained equilibria
of set-valued maps. The abstract results are applied to show existence of
(strong) steady state solutions to some weakly coupled systems of drift
reaction-diffusion equations or differential inclusions of this type. In
particular we get the existence of strong solutions to the Dirichlet, Neumann
and periodic boundary problems for elliptic partial differential inclusions
under the presence of state constraints of different type. Certain aspects of
the Bernstein theory for bvp for second order ODE are studied, too. No
assumptions concerning structural coupling (monotonicity, cooperativity) are
undertaken.

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The categories of this paper are: stat
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: plasm-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph