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14
Variational aspects of Laplace eigenvalues on Riemannian surfaces
 Adv. Math
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The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 8 (6 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 8 (5 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
Critical Sets of Smooth SOLUTIONS TO ELLIPTIC EQUATIONS in Dimension 3
, 2001
"... Let u ̸ ≡ const satisfy an elliptic equation L 0u ≡ Σai,jDiju + ΣbjDju = 0 with smooth coefficients in a domain in R 3. It is shown that the critical set ∇u  −1 {0} has locally finite 1dimensional Hausdorff measure. This implies in particular that for a solution u ̸ ≡ 0 of (L0 + c)u = 0, with c ..."
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Cited by 5 (0 self)
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Let u ̸ ≡ const satisfy an elliptic equation L 0u ≡ Σai,jDiju + ΣbjDju = 0 with smooth coefficients in a domain in R 3. It is shown that the critical set ∇u  −1 {0} has locally finite 1dimensional Hausdorff measure. This implies in particular that for a solution u ̸ ≡ 0 of (L0 + c)u = 0, with c ∈ C ∞ , the critical zero set u −1 {0} ∩ ∇u  −1 {0} has locally finite 1dimensional Hausdorff measure.
On Serrin’s symmetry result in nonsmooth domains and its applications, preprint
"... The goal of this paper is to show that Serrin’s result on overdetermined problems holds true for symmetric nonsmooth domains. Specifically, we show that if a nonsmooth domain D satisfies appropriate symmetry and convexity assumptions, and there exists a positive solution to a general overdetermine ..."
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The goal of this paper is to show that Serrin’s result on overdetermined problems holds true for symmetric nonsmooth domains. Specifically, we show that if a nonsmooth domain D satisfies appropriate symmetry and convexity assumptions, and there exists a positive solution to a general overdetermined problem on D, then D must be a ball. As an application, we improve results on symmetry of nonnegative solutions of Dirichlet problems. 1
unknown title
, 2009
"... Nodal sets of magnetic Schrödinger operators of Aharonov–Bohm type and energy minimizing partitions ∗ ..."
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Nodal sets of magnetic Schrödinger operators of Aharonov–Bohm type and energy minimizing partitions ∗
unknown title
, 2009
"... Nodal sets of magnetic Schrödinger operators of Aharonov–Bohm type and energy minimizing partitions ∗ ..."
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Nodal sets of magnetic Schrödinger operators of Aharonov–Bohm type and energy minimizing partitions ∗