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14
BERNSTEIN AND DE GIORGI TYPE PROBLEMS: NEW RESULTS VIA A GEOMETRIC APPROACH
"... Abstract. We use a Poincaré type formula and level set analysis to detect onedimensional symmetry of stable solutions of possibly degenerate or singular elliptic equation of the form div a(∇u(x))∇u(x) + f(u(x)) = 0. Our setting is very general and, as particular cases, we obtain new proofs of a ..."
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Cited by 9 (5 self)
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Abstract. We use a Poincaré type formula and level set analysis to detect onedimensional symmetry of stable solutions of possibly degenerate or singular elliptic equation of the form div a(∇u(x))∇u(x) + f(u(x)) = 0. Our setting is very general and, as particular cases, we obtain new proofs of a conjecture of De Giorgi for phase transitions in R 2 and R 3 and of the Bernstein problem on the flatness of minimal area graphs in R 3. A onedimensional symmetry result in the halfspace is also obtained as a byproduct of our analysis. Our approach is also flexible to nonelliptic operators: as an application, we prove onedimensional symmetry for 1Laplacian type operators. 1.
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 7 (4 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Remarks On Virtual Bound States For SemiBounded Operators
 laptev@math.kth.se T. Weidl: Department of Mathematics, Royal Institute of Technology, S  100 44
, 1999
"... . We calculate the number of bound states appearing below the spectrum of a semibounded operator in the case of a weak, nonsigndefined perturbation. The abstract result generalizes the BirmanSchwinger principle to this case. We discuss a number of examples, in particular higher order differentia ..."
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Cited by 4 (1 self)
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. We calculate the number of bound states appearing below the spectrum of a semibounded operator in the case of a weak, nonsigndefined perturbation. The abstract result generalizes the BirmanSchwinger principle to this case. We discuss a number of examples, in particular higher order differential operators, critical Schrodinger operators, systems of second order differential operators, Schrodinger type operators with magnetic fields and the twodimensional Pauli operator with a localized magnetic field. 1. Introduction Let A be a selfadjoint semibounded operator on some separable Hilbert space H with the lower bound zero. Let V be a suitable hermitian perturbation, such that the operator A(ff) := A \Gamma ffV is welldefined and selfadjoint for ff 2 (0; ff 0 ); ff 0 ? 0. Assume moreover, that the negative spectrum of A(ff) consists only of a finite number n \Gamma (A(ff); 0) of negative eigenvalues for all ff 2 (0; ff 0 ). Clearly these eigenvalues will tend to zero as ff ! ...
I.: Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
"... Abstract. 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 4 (3 self)
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Abstract. 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.
RIGIDITY RESULTS FOR SOME BOUNDARY QUASILINEAR PHASE TRANSITIONS
, 803
"... Abstract. We consider a quasilinear equation given in the halfspace, i.e. a so called boundary reaction problem. Our concerns are a geometric Poincaré inequality and, as a byproduct of this inequality, a result on the symmetry of lowdimensional bounded stable solutions, under some suitable assumpt ..."
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Cited by 1 (0 self)
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Abstract. We consider a quasilinear equation given in the halfspace, i.e. a so called boundary reaction problem. Our concerns are a geometric Poincaré inequality and, as a byproduct of this inequality, a result on the symmetry of lowdimensional bounded stable solutions, under some suitable assumptions on the nonlinearities. More precisely, we analyze the following boundary problem −div (a(x, ∇u)∇u) + g(x, u) = 0 on R
LARGE TIME BEHAVIOR OF THE HEAT KERNEL OF TWODIMENSIONAL MAGNETIC SCHRÖDINGER OPERATORS
, 2010
"... We study the heat semigroup generated by twodimensional Schrödinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some ondiagonal heat kernel est ..."
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We study the heat semigroup generated by twodimensional Schrödinger operators with compactly supported magnetic field. We show that if the field is radial, then the large time behavior of the associated heat kernel is determined by its total flux. We also establish some ondiagonal heat kernel estimates and discuss their applications for solutions to the heat equation. An exact formula for the heat kernel, and for its large time asymptotic, is derived in the case of the AharonovBohm magnetic field.
Fibered nonlinearities for p(x)Laplace equations
, 2008
"... Abstract. In R m × R n−m, endowed with coordinates X = (x, y), we consider the PDE −div ` α(x)∇u(X)  p(x)−2 ∇u(X) ´ = f(x, u(X)). We prove a geometric inequality and a symmetry result. EV is supported by MIUR, project “Variational methods and Nonlinear Differential Equations”. 1 ..."
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Abstract. In R m × R n−m, endowed with coordinates X = (x, y), we consider the PDE −div ` α(x)∇u(X)  p(x)−2 ∇u(X) ´ = f(x, u(X)). We prove a geometric inequality and a symmetry result. EV is supported by MIUR, project “Variational methods and Nonlinear Differential Equations”. 1
for Strongly Local Dirichlet Forms Dedicated to Jürgen Voigt in celebration of his 65th birthday
, 2008
"... Communicated by Heinz Siedentop Abstract. The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. 2000 Mathematics Subject Classification: 35P05, 81Q10 ..."
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Communicated by Heinz Siedentop Abstract. The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. 2000 Mathematics Subject Classification: 35P05, 81Q10
A2. The class Kv
"... ABSTRACT. Let H = \L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and regularity, and in particular allow V which are unbounded below. We give a general survey of the p ..."
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ABSTRACT. Let H = \L + V be a general Schrödinger operator on R" (v~> 1), where A is the Laplace differential operator and V is a potential function on which we assume minimal hypotheses of growth and regularity, and in particular allow V which are unbounded below. We give a general survey of the properties of e ~ tH, t> 0, and related mappings given in terms of solutions of initial value problems for the differential equation du/dt + Hu = 0. Among the subjects treated are L ^properties of these maps, existence of continuous integral kernels for them, and regularity properties of eigenfunctions, including Harnack's inequality.