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Boundary behavior of harmonic functions for truncated stable processes
 J. THEORET. PROBAB
, 2008
"... For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, 2007) we have studied the potential theory of trunca ..."
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Cited by 26 (16 self)
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For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1{x<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, 2007) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in nonconvex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in R d called bounded roughly connected κfat open sets (including bounded nonconvex κfat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κfat open sets.
Spectral stability of the Neumann Laplacian
 J. Diff. Eq
"... We prove the equivalence of Hardy and Sobolevtype inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of th ..."
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Cited by 3 (0 self)
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We prove the equivalence of Hardy and Sobolevtype inequalities, certain uniform bounds on the heat kernel and some spectral regularity properties of the Neumann Laplacian associated with an arbitrary region of finite measure in Euclidean space. We also prove that if one perturbs the boundary of the region within a uniform Hölder category then the eigenvalues of the Neumann Laplacian change by a small and explicitly estimated amount.
and
, 2006
"... For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1 {x<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other ..."
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For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1 {x<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of a truncated symmetric stable process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in nonconvex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in R d called bounded roughly connected κfat open sets (including bounded nonconvex κfat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κfat open sets.
and
, 2007
"... For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1 {x<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other ..."
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For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1 {x<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in nonconvex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in R d called bounded roughly connected κfat open sets (including bounded nonconvex κfat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κfat open sets.
and
, 2006
"... For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1 {x<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other ..."
Abstract
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For any α ∈ (0, 2), a truncated symmetric αstable process in R d is a symmetric Lévy process in R d with no diffusion part and with a Lévy density given by cx  −d−α 1 {x<1} for some constant c. In [24] we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of a truncated symmetric stable process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in nonconvex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in R d called bounded roughly connected κfat open sets (including bounded nonconvex κfat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κfat open sets.
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2 On boundary behavior and Dirichlet problem for Beltrami equations
, 2013
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