Xiao Zhong - Academia.edu (original) (raw)

Papers by Xiao Zhong

The Michigan Mathematical Journal

We establish an essentially sharp modulus of continuity for mappings of subexponentially integrab... more We establish an essentially sharp modulus of continuity for mappings of subexponentially integrable distortion.

Proceedings of the American Mathematical Society

We show that sets of n − p + α(p − 1) Hausdorff measure zero are re-movable for α-Hölder continuo... more We show that sets of n − p + α(p − 1) Hausdorff measure zero are re-movable for α-Hölder continuous solutions to quasilinear elliptic equations similar to the p-Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.

St Petersburg Mathematical Journal

Anisotropic variational integrals of (p,q)-growth are considered. For the scalar case, the interi... more Anisotropic variational integrals of (p,q)-growth are considered. For the scalar case, the interior C 1,α -regularity of bounded local minimizers is proved under the assumption that q≤2p, and a famous counterexample of Giaquinta is discussed. In the vector case, some higher integrability result for the gradient is obtained.

St Petersburg Mathematical Journal

The authors consider the following extension of the stationary Navier-Stokes equations -div(T(·,ε... more The authors consider the following extension of the stationary Navier-Stokes equations -div(T(·,ε(u)))+∇π+[∇u]u=g, posed in a bounded and Lipschitz domain Ω of ℝ n , n=2 or 3. Here ε(u) is the symmetric gradient of u and T is the gradient with respect to its second argument of a potential f(x,ε) which satisfies some growth conditions on its second order partial derivatives D ε 2 and D x D ε . The volumic forces g are supposed to belong to L ∞ (Ω,ℝ n ). Homogeneous Dirichlet boundary conditions are imposed on the boundary ∂Ω. The main result proves the existence of a solution (v,π)∈W o p 1 ∩W t,loc 2 (Ω,ℝ n )×W s,loc 1 (Ω), where t and s, greater than 1, are linked to the above growth conditions, which involve p and some q ¯. This result is proved assuming that p is large enough and q ¯ is small enough. For the proof, the authors use their previous results which consider various situations for f. The key argument is the proof of some estimates on the weak solution (in a variational f...

Annales- Academiae Scientiarum Fennicae Mathematica

We use a scaling or blow up argument to obtain estimates to solutions of equations of p-Laplacian... more We use a scaling or blow up argument to obtain estimates to solutions of equations of p-Laplacian type.

We prove an estimate of the growth of a nonnegative A -subharmonic function in R n in terms of th... more We prove an estimate of the growth of a nonnegative A -subharmonic function in R n in terms of the Wolfi potential of its Riesz measure. Our estimate can be viewed as a counterpart to Nevanlinna's flrst fundamental theorem for subharmonic functions in the nonlinear setting. As a consequence, we prove that a nonnegative A -subharmonic function has the same

We study the removable singularities for solutions to the Beltrami equation @f = µ@f, assuming th... more We study the removable singularities for solutions to the Beltrami equation @f = µ@f, assuming that the coecient µ lies on some Sobolev space W1,p, p • 2. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported

Manuscripta Mathematica, 2005

In this paper we prove a lemma on the higher integrability of functions and discuss its applicati... more In this paper we prove a lemma on the higher integrability of functions and discuss its applications to the regularity theory of two-dimensional generalized Newtonian fluids.

Starting from Giaquinta's counterexample [Gi] we introduce the class of splitting functionals bei... more Starting from Giaquinta's counterexample [Gi] we introduce the class of splitting functionals being of (p, q)-growth with exponents p ≤ q < ∞ and show for the scalar case that locally bounded local minimizers are of class C 1,µ . Note that to our knowledge the only C 1,µ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini's paper [Ma1], Theorem A, and later on in the work of Fusco and Sbordone [FS], Theorem 4.2.

SIAM Journal on Mathematical Analysis, 2012

ABSTRACT We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplac... more ABSTRACT We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map. The result is constructive and local, and gives a method for determining the coefficient at a boundary point from measurements in a small neighborhood. The proofs work with the nonlinear equation directly instead of being based on linearization. In the complex valued case we employ complex geometrical optics type solutions based on p-harmonic exponentials, while for the real case we use p-harmonic functions first introduced by Wolff.

Abstract. We prove an estimate of the growth of a nonnegative A-subharmonic function in R, in ter... more Abstract. We prove an estimate of the growth of a nonnegative A-subharmonic function in R, in terms of the Wolfi potential of its Riesz measure. Our estimate can be viewed as a counterpart to Nevanlinna's 炉rst fundamental theorem for subharmonic functions in the nonlinear setting. As a consequence, we prove that a nonnegative A-subharmonic function has the same order as the Wolfi potential of its Riesz measure.

Revista Matemática Iberoamericana, 2000

We establish continuity, openness and discreteness, and the condition (N ) for mappings of finite... more We establish continuity, openness and discreteness, and the condition (N ) for mappings of finite distortion under minimal integrability assumptions on the distortion. 2000 Mathematics Subject Classification: 30C65.

Publicacions Matemàtiques, 2009

We study the removable singularities for solutions to the Beltrami equation ∂f = µ ∂f , assuming ... more We study the removable singularities for solutions to the Beltrami equation ∂f = µ ∂f , assuming that the coefficient µ lies on some Sobolev space W 1,p , p ≤ 2. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient µ ∈ W 1,2 preserve compact sets of σ-finite length and vanishing analytic capacity, even though they need not be bilipschitz.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2008

ABSTRACT We give a new and elementary proof of the known result: a non-constant mapping of finite... more ABSTRACT We give a new and elementary proof of the known result: a non-constant mapping of finite distortion f : Ω n → n is discrete and open, provided that its distortion function if n = 2 and that for some p &gt; n − 1 if n ≥ 3.(Received July 31 2006)(Accepted October 11 2007)

Potential Analysis, 2007

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions... more We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C 1,1 -domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. 2000 Mathematics Subject Classification. 35J65, 35J25, 31C45, 31B25, 31B05. Harnack principle has garnered a lot of attention, and many papers have been devoted to the study of the boundary Harnack principle not only for Euclidean domains but also for manifolds. Among them, Caffarelli-Fabes-Mortola-Salsa [11], Jerison-Kenig [17] and Bass-Burdzy-Bañuelos [9, 8] gave significant extensions for the boundary Harnack principle for Euclidean domains. The boundary Harnack principle is now well-known for Lipschitz domains, non-tangentially accessible (NTA) domains, and more generally for Hölder domains. In [1], the first named author showed that a uniform domain satisfies a local version of the boundary Harnack principle which is stronger than the above boundary Harnack principle. It has applications to the study of the Martin boundary. In particular, this version of the boundary Harnack principle shows that the Martin boundary and the Euclidean boundary of a uniform domain are homeomorphic, a fact well-known for a Lipschitz domain and an NTA domain.

Journal of Mathematical Sciences, 2011

ABSTRACT We show that a velocity field u satisfying the stationary Navier–Stokes equations on the... more ABSTRACT We show that a velocity field u satisfying the stationary Navier–Stokes equations on the entire plane must be constant under the growth condition lim sup |x|−α |u(x)| &lt; ∞ as |x| → ∞ for some α ∈ [0, 1/7).† Bibliography: 10 titles.

We consider the simplest form of a second order, linear, degenerate, divergence structure equatio... more We consider the simplest form of a second order, linear, degenerate, divergence structure equation in the plane. Under an integrability condition on the degenerate function, we prove that the solutions are continuous.

Arkiv för Matematik, 2008

In this note, we consider the regularity of solutions of the nolinear elliptic systems of n-Lapla... more In this note, we consider the regularity of solutions of the nolinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space L n,∞ . We also obtain the a priori global and local estimates for the L n,∞ -norm of the gradients of the solutions without using BM O-estimates. The proofs are based on a new lemma on the higher integrabilty of functions.

The Michigan Mathematical Journal

We establish an essentially sharp modulus of continuity for mappings of subexponentially integrab... more We establish an essentially sharp modulus of continuity for mappings of subexponentially integrable distortion.

Proceedings of the American Mathematical Society

We show that sets of n − p + α(p − 1) Hausdorff measure zero are re-movable for α-Hölder continuo... more We show that sets of n − p + α(p − 1) Hausdorff measure zero are re-movable for α-Hölder continuous solutions to quasilinear elliptic equations similar to the p-Laplacian. The result is optimal. We also treat larger sets in terms of a growth condition. In particular, our results apply to quasiregular mappings.

St Petersburg Mathematical Journal

Anisotropic variational integrals of (p,q)-growth are considered. For the scalar case, the interi... more Anisotropic variational integrals of (p,q)-growth are considered. For the scalar case, the interior C 1,α -regularity of bounded local minimizers is proved under the assumption that q≤2p, and a famous counterexample of Giaquinta is discussed. In the vector case, some higher integrability result for the gradient is obtained.

St Petersburg Mathematical Journal

The authors consider the following extension of the stationary Navier-Stokes equations -div(T(·,ε... more The authors consider the following extension of the stationary Navier-Stokes equations -div(T(·,ε(u)))+∇π+[∇u]u=g, posed in a bounded and Lipschitz domain Ω of ℝ n , n=2 or 3. Here ε(u) is the symmetric gradient of u and T is the gradient with respect to its second argument of a potential f(x,ε) which satisfies some growth conditions on its second order partial derivatives D ε 2 and D x D ε . The volumic forces g are supposed to belong to L ∞ (Ω,ℝ n ). Homogeneous Dirichlet boundary conditions are imposed on the boundary ∂Ω. The main result proves the existence of a solution (v,π)∈W o p 1 ∩W t,loc 2 (Ω,ℝ n )×W s,loc 1 (Ω), where t and s, greater than 1, are linked to the above growth conditions, which involve p and some q ¯. This result is proved assuming that p is large enough and q ¯ is small enough. For the proof, the authors use their previous results which consider various situations for f. The key argument is the proof of some estimates on the weak solution (in a variational f...

Annales- Academiae Scientiarum Fennicae Mathematica

We use a scaling or blow up argument to obtain estimates to solutions of equations of p-Laplacian... more We use a scaling or blow up argument to obtain estimates to solutions of equations of p-Laplacian type.

We prove an estimate of the growth of a nonnegative A -subharmonic function in R n in terms of th... more We prove an estimate of the growth of a nonnegative A -subharmonic function in R n in terms of the Wolfi potential of its Riesz measure. Our estimate can be viewed as a counterpart to Nevanlinna's flrst fundamental theorem for subharmonic functions in the nonlinear setting. As a consequence, we prove that a nonnegative A -subharmonic function has the same

We study the removable singularities for solutions to the Beltrami equation @f = µ@f, assuming th... more We study the removable singularities for solutions to the Beltrami equation @f = µ@f, assuming that the coecient µ lies on some Sobolev space W1,p, p • 2. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported

Manuscripta Mathematica, 2005

In this paper we prove a lemma on the higher integrability of functions and discuss its applicati... more In this paper we prove a lemma on the higher integrability of functions and discuss its applications to the regularity theory of two-dimensional generalized Newtonian fluids.

Starting from Giaquinta's counterexample [Gi] we introduce the class of splitting functionals bei... more Starting from Giaquinta's counterexample [Gi] we introduce the class of splitting functionals being of (p, q)-growth with exponents p ≤ q < ∞ and show for the scalar case that locally bounded local minimizers are of class C 1,µ . Note that to our knowledge the only C 1,µ -results without imposing a relation between p and q concern the case of two independent variables as it is outlined in Marcellini's paper [Ma1], Theorem A, and later on in the work of Fusco and Sbordone [FS], Theorem 4.2.

SIAM Journal on Mathematical Analysis, 2012

ABSTRACT We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplac... more ABSTRACT We study an inverse problem for nonlinear elliptic equations modelled after the p-Laplacian. It is proved that the boundary values of a conductivity coefficient are uniquely determined from boundary measurements given by a nonlinear Dirichlet-to-Neumann map. The result is constructive and local, and gives a method for determining the coefficient at a boundary point from measurements in a small neighborhood. The proofs work with the nonlinear equation directly instead of being based on linearization. In the complex valued case we employ complex geometrical optics type solutions based on p-harmonic exponentials, while for the real case we use p-harmonic functions first introduced by Wolff.

Abstract. We prove an estimate of the growth of a nonnegative A-subharmonic function in R, in ter... more Abstract. We prove an estimate of the growth of a nonnegative A-subharmonic function in R, in terms of the Wolfi potential of its Riesz measure. Our estimate can be viewed as a counterpart to Nevanlinna's 炉rst fundamental theorem for subharmonic functions in the nonlinear setting. As a consequence, we prove that a nonnegative A-subharmonic function has the same order as the Wolfi potential of its Riesz measure.

Revista Matemática Iberoamericana, 2000

We establish continuity, openness and discreteness, and the condition (N ) for mappings of finite... more We establish continuity, openness and discreteness, and the condition (N ) for mappings of finite distortion under minimal integrability assumptions on the distortion. 2000 Mathematics Subject Classification: 30C65.

Publicacions Matemàtiques, 2009

We study the removable singularities for solutions to the Beltrami equation ∂f = µ ∂f , assuming ... more We study the removable singularities for solutions to the Beltrami equation ∂f = µ ∂f , assuming that the coefficient µ lies on some Sobolev space W 1,p , p ≤ 2. Our results are based on an extended version of the well known Weyl's lemma, asserting that distributional solutions are actually true solutions. Our main result is that quasiconformal mappings with compactly supported Beltrami coefficient µ ∈ W 1,2 preserve compact sets of σ-finite length and vanishing analytic capacity, even though they need not be bilipschitz.

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2008

ABSTRACT We give a new and elementary proof of the known result: a non-constant mapping of finite... more ABSTRACT We give a new and elementary proof of the known result: a non-constant mapping of finite distortion f : Ω n → n is discrete and open, provided that its distortion function if n = 2 and that for some p &gt; n − 1 if n ≥ 3.(Received July 31 2006)(Accepted October 11 2007)

Potential Analysis, 2007

We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions... more We establish a scale-invariant version of the boundary Harnack principle for p-harmonic functions in Euclidean C 1,1 -domains and obtain estimates for the decay rates of positive p-harmonic functions vanishing on a segment of the boundary in terms of the distance to the boundary. We use these estimates to study the behavior of conformal Martin kernel functions and positive p-superharmonic functions near the boundary of the domain. 2000 Mathematics Subject Classification. 35J65, 35J25, 31C45, 31B25, 31B05. Harnack principle has garnered a lot of attention, and many papers have been devoted to the study of the boundary Harnack principle not only for Euclidean domains but also for manifolds. Among them, Caffarelli-Fabes-Mortola-Salsa [11], Jerison-Kenig [17] and Bass-Burdzy-Bañuelos [9, 8] gave significant extensions for the boundary Harnack principle for Euclidean domains. The boundary Harnack principle is now well-known for Lipschitz domains, non-tangentially accessible (NTA) domains, and more generally for Hölder domains. In [1], the first named author showed that a uniform domain satisfies a local version of the boundary Harnack principle which is stronger than the above boundary Harnack principle. It has applications to the study of the Martin boundary. In particular, this version of the boundary Harnack principle shows that the Martin boundary and the Euclidean boundary of a uniform domain are homeomorphic, a fact well-known for a Lipschitz domain and an NTA domain.

Journal of Mathematical Sciences, 2011

ABSTRACT We show that a velocity field u satisfying the stationary Navier–Stokes equations on the... more ABSTRACT We show that a velocity field u satisfying the stationary Navier–Stokes equations on the entire plane must be constant under the growth condition lim sup |x|−α |u(x)| &lt; ∞ as |x| → ∞ for some α ∈ [0, 1/7).† Bibliography: 10 titles.

We consider the simplest form of a second order, linear, degenerate, divergence structure equatio... more We consider the simplest form of a second order, linear, degenerate, divergence structure equation in the plane. Under an integrability condition on the degenerate function, we prove that the solutions are continuous.

Arkiv för Matematik, 2008

In this note, we consider the regularity of solutions of the nolinear elliptic systems of n-Lapla... more In this note, we consider the regularity of solutions of the nolinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space L n,∞ . We also obtain the a priori global and local estimates for the L n,∞ -norm of the gradients of the solutions without using BM O-estimates. The proofs are based on a new lemma on the higher integrabilty of functions.