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Search: MSC category 35J ( Elliptic equations and systems [See also 58J10, 58J20] )

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1. CJM Online first

Jiang, Feida; Trudinger, Neil S; Xiang, Ni
On the Neumann problem for Monge-Ampère type equations
In this paper, we study the global regularity for regular Monge-Ampère type equations associated with semilinear Neumann boundary conditions. By establishing a priori estimates for second order derivatives, the classical solvability of the Neumann boundary value problem is proved under natural conditions. The techniques build upon the delicate and intricate treatment of the standard Monge-Ampère case by Lions, Trudinger and Urbas in 1986 and the recent barrier constructions and second derivative bounds by Jiang, Trudinger and Yang for the Dirichlet problem. We also consider more general oblique boundary value problems in the strictly regular case.

Keywords:semilinear Neumann problem, Monge-Ampère type equation, second derivative estimates
Categories:35J66, 35J96

2. CJM Online first

Emamizadeh, Behrouz; Farjudian, Amin; Zivari-Rezapour, Mohsen
Optimization related to some nonlocal problems of Kirchhoff type
In this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton we are able to show that both problems are solvable, and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions. The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type, which is stable. Some numerical results are included to confirm the theory.

Keywords:Kirchhoff equation, rearrangements of functions, maximization, existence, optimality condition
Categories:35J20, 35J25

3. CJM 2015 (vol 67 pp. 1161)

Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun
Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\ge 2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$, the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$ is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$.

Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform
Categories:42B30, 42B35, 35J70

4. CJM 2013 (vol 65 pp. 1217)

Cruz, Victor; Mateu, Joan; Orobitg, Joan
Beltrami Equation with Coefficient in Sobolev and Besov Spaces
Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.

Keywords:quasiregular mappings, Beltrami equation, Sobolev spaces, Calderón-Zygmund operators
Categories:30C62, 35J99, 42B20

5. CJM 2013 (vol 66 pp. 641)

Grigor'yan, Alexander; Hu, Jiaxin
Heat Kernels and Green Functions on Metric Measure Spaces
We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling propety, the elliptic Harnack inequality and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball, that uses two-sided estimates of a Green function in a ball.

Keywords:Dirichlet form, heat kernel, Green function, capacity
Categories:35K08, 28A80, 31B05, 35J08, 46E35, 47D07

6. CJM 2012 (vol 65 pp. 927)

Wang, Liping; Zhao, Chunyi
Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$
We consider the following prescribed boundary mean curvature problem in $ \mathbb B^N$ with the Euclidean metric: \[ \begin{cases} \displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N, \\[2ex] \displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1}, \end{cases} \] where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$. We show that if $\widetilde K(x)$ has a local maximum point, then the above problem has infinitely many positive solutions that are not rotationally symmetric on $\mathbb S^{N-1}$.

Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reduction
Categories:35J25, 35J65, 35J67

7. CJM 2012 (vol 64 pp. 1395)

Rodney, Scott
Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems With Rough Coefficients
This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form \begin{align*} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta \\ u&=\varphi\text{ on }\partial \Theta. \end{align*} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in previous works. Sawyer and Wheeden give a regularity theory for a subset of the class of equations dealt with here.

Keywords:degenerate quadratic forms, linear equations, rough coefficients, subelliptic, weak solutions
Categories:35A01, 35A02, 35D30, 35J70, 35H20

8. CJM 2012 (vol 65 pp. 702)

Taylor, Michael
Regularity of Standing Waves on Lipschitz Domains
We analyze the regularity of standing wave solutions to nonlinear Schrödinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in Hölder spaces.

Keywords:standing waves, elliptic regularity, Lipschitz domain
Categories:35J25, 35J65

9. CJM 2011 (vol 64 pp. 924)

McCann, Robert J.; Pass, Brendan; Warren, Micah
Rectifiability of Optimal Transportation Plans
The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

Categories:49K20, 49K60, 35J96, 58C07

10. CJM 2011 (vol 64 pp. 217)

Tang, Lin
$W_\omega^{2,p}$-Solvability of the Cauchy-Dirichlet Problem for Nondivergence Parabolic Equations with BMO Coefficients
In this paper, we establish the regularity of strong solutions to nondivergence parabolic equations with BMO coefficients in nondoubling weighted spaces.

Categories:35J45, 35J55

11. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps
We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identities
Categories:28A80, , , , 35P20, 35J05, 43A05, 47A75

12. CJM 2010 (vol 62 pp. 808)

Legendre, Eveline
Extrema of Low Eigenvalues of the Dirichlet-Neumann Laplacian on a Disk
We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet--Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

Keywords: Laplacian, eigenvalues, Dirichlet-Neumann mixed boundary condition, Zaremba's problem
Categories:35J25, 35P15

13. CJM 2009 (vol 62 pp. 19)

Bouchekif, Mohammed; Nasri, Yasmina
Solutions for Semilinear Elliptic Systems with Critical Sobolev Exponent and Hardy Potential
In this paper we consider an elliptic system with an inverse square potential and critical Sobolev exponent in a bounded domain of $\mathbb{R}^N$. By variational methods we study the existence results.

Keywords:critical Sobolev exponent, Palais--Smale condition, Linking theorem, Hardy potential
Categories:35B25, 35B33, 35J50, 35J60

14. CJM 2008 (vol 60 pp. 822)

Kuwae, Kazuhiro
Maximum Principles for Subharmonic Functions Via Local Semi-Dirichlet Forms
Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Keywords:positivity preserving form, semi-Dirichlet form, Dirichlet form, subharmonic functions, superharmonic functions, harmonic functions, weak maximum principle, strong maximum principle, irreducibility, absolute continuity condition
Categories:31C25, 35B50, 60J45, 35J, 53C, 58

15. CJM 2007 (vol 59 pp. 943)

Finster, Felix; Kraus, Margarita
A Weighted $L^2$-Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds
We derive a weighted $L^2$-estimate of the Witten spinor in a complete Riemannian spin manifold~$(M^n, g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of~$M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.

Categories:83C60, 35Q75, 35J45, 58J05

16. CJM 2007 (vol 59 pp. 742)

Gil, Juan B.; Krainer, Thomas; Mendoza, Gerardo A.
Geometry and Spectra of Closed Extensions of Elliptic Cone Operators
We study the geometry of the set of closed extensions of index $0$ of an elliptic differential cone operator and its model operator in connection with the spectra of the extensions, and we give a necessary and sufficient condition for the existence of rays of minimal growth for such operators.

Keywords:resolvents, manifolds with conical singularities, spectral theor, boundary value problems, Grassmannians
Categories:58J50, 35J70, 14M15

17. CJM 2006 (vol 58 pp. 64)

Filippakis, Michael; Gasiński, Leszek; Papageorgiou, Nikolaos S.
Multiplicity Results for Nonlinear Neumann Problems
In this paper we study nonlinear elliptic problems of Neumann type driven by the $p$-Laplac\-ian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a $C^1$-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-$p$-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative $p$-Laplacian with Neumann boundary condition.

Keywords:Nonsmooth critical point theory, locally Lipschitz function,, Clarke subdifferential, Neumann problem, strong resonance,, second deformation theorem, nonsmooth symmetric mountain pass theorem,, $p$-Laplacian
Categories:35J20, 35J60, 35J85

18. CJM 2005 (vol 57 pp. 771)

Schrohe, E.; Seiler, J.
The Resolvent of Closed Extensions of Cone Differential Operators
We study closed extensions $\underline A$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted $L_p$-space. Under suitable conditions we show that the resolvent $(\lambda-\underline A)^{-1}$ exists in a sector of the complex plane and decays like $1/|\lambda|$ as $|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underline A$. As an application we treat the Laplace--Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$.

Keywords:Manifolds with conical singularities, resolvent, maximal regularity
Categories:35J70, 47A10, 58J40

19. CJM 2004 (vol 56 pp. 655)

Tao, Xiangxing; Wang, Henggeng
On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains
We consider the Neumann problem for the Schr\"odinger equations $-\Delta u+Vu=0$, with singular nonnegative potentials $V$ belonging to the reverse H\"older class $\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given boundary data $g$ in $H^p$ or $L^p$ for $1-\epsilon
Keywords:Neumann problem, Schrödinger equation, Lipschitz, domain, reverse Hölder class, $H^p$ space
Categories:42B20, 35J10

20. CJM 2002 (vol 54 pp. 1121)

Bao, Jiguang
Fully Nonlinear Elliptic Equations on General Domains
By means of the Pucci operator, we construct a function $u_0$, which plays an essential role in our considerations, and give the existence and regularity theorems for the bounded viscosity solutions of the generalized Dirichlet problems of second order fully nonlinear elliptic equations on the general bounded domains, which may be irregular. The approximation method, the accretive operator technique and the Caffarelli's perturbation theory are used.

Keywords:Pucci operator, viscosity solution, existence, $C^{2,\psi}$ regularity, Dini condition, fully nonlinear equation, general domain, accretive operator, approximation lemma
Categories:35D05, 35D10, 35J60, 35J67

21. CJM 2002 (vol 54 pp. 945)

Boivin, André; Gauthier, Paul M.; Paramonov, Petr V.
Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications
Given a homogeneous elliptic partial differential operator $L$ with constant complex coefficients and a class of functions (jet-distributions) which are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and which belong locally to a Banach space $V$, we consider the problem of approximating in the norm of $V$ the functions in this class by ``analytic'' and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth, Arakelyan (including tangential) and Carleman type theorems for a large class of Banach spaces $V$ and operators $L$. Important applications to boundary value problems of solutions of homogeneous elliptic partial differential equations are obtained, including the solution of a generalized Dirichlet problem.

Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$-meromorphic and $L$-analytic functions, localization operator, Banach space of distributions, Dirichlet problem
Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30

22. CJM 2000 (vol 52 pp. 757)

Hanani, Abdellah
Le problème de Neumann pour certaines équations du type de Monge-Ampère sur une variété riemannienne
Let $(M_n,g)$ be a strictly convex riemannian manifold with $C^{\infty}$ boundary. We prove the existence\break of classical solution for the nonlinear elliptic partial differential equation of Monge-Amp\`ere:\break $\det (-u\delta^i_j + \nabla^i_ju) = F(x,\nabla u;u)$ in $M$ with a Neumann condition on the boundary of the form $\frac{\partial u}{\partial \nu} = \varphi (x,u)$, where $F \in C^{\infty} (TM \times \bbR)$ is an everywhere strictly positive function satisfying some assumptions, $\nu$ stands for the unit normal vector field and $\varphi \in C^{\infty} (\partial M \times \bbR)$ is a non-decreasing function in $u$.

Keywords:connexion de Levi-Civita, équations de Monge-Ampère, problème de Neumann, estimées a priori, méthode de continuité
Categories:35J60, 53C55, 58G30

23. CJM 2000 (vol 52 pp. 522)

Gui, Changfeng; Wei, Juncheng
On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems
We consider the problem \begin{equation*} \begin{cases} \varepsilon^2 \Delta u - u + f(u) = 0, u > 0 & \mbox{in } \Omega\\ \frac{\partial u}{\partial \nu} = 0 & \mbox{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $R^N$, $\ve>0$ is a small parameter and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as $\epsilon$ approaches zero, at multiple critical points of the mean curvature function $H(P)$, $P \in \partial \Omega$. It is also proved that this equation has multiple interior spike solutions which concentrate, as $\ep\to 0$, at {\it sphere packing\/} points in $\Om$. In this paper, we prove the existence of solutions with multiple spikes {\it both\/} on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

Keywords:mixed multiple spikes, nonlinear elliptic equations
Categories:35B40, 35B45, 35J40

24. CJM 1998 (vol 50 pp. 487)

Barlow, Martin T.
On the Liouville property for divergence form operators
In this paper we construct a bounded strictly positive function $\sigma$ such that the Liouville property fails for the divergence form operator $L=\nabla (\sigma^2 \nabla)$. Since in addition $\Delta \sigma/\sigma$ is bounded, this example also gives a negative answer to a problem of Berestycki, Caffarelli and Nirenberg concerning linear Schr\"odinger operators.

Categories:31C05, 60H10, 35J10

25. CJM 1998 (vol 50 pp. 40)

Engliš, Miroslav; Peetre, Jaak
Green's functions for powers of the invariant Laplacian
The aim of the present paper is the computation of Green's functions for the powers $\DDelta^m$ of the invariant Laplace operator on rank-one Hermitian symmetric spaces. Starting with the noncompact case, the unit ball in $\CC^d$, we obtain a complete result for $m=1,2$ in all dimensions. For $m\ge3$ the formulas grow quite complicated so we restrict ourselves to the case of the unit disc ($d=1$) where we develop a method, possibly applicable also in other situations, for reducing the number of integrations by half, and use it to give a description of the boundary behaviour of these Green functions and to obtain their (multi-valued) analytic continuation to the entire complex plane. Next we discuss the type of special functions that turn up (hyperlogarithms of Kummer). Finally we treat also the compact case of the complex projective space $\Bbb P^d$ (for $d=1$, the Riemann sphere) and, as an application of our results, use eigenfunction expansions to obtain some new identities involving sums of Legendre ($d=1$) or Jacobi ($d>1$) polynomials and the polylogarithm function. The case of Green's functions of powers of weighted (no longer invariant, but only covariant) Laplacians is also briefly discussed.

Keywords:Invariant Laplacian, Green's functions, dilogarithm, trilogarithm, Legendre and Jacobi polynomials, hyperlogarithms
Categories:35C05, 33E30, 33C45, 34B27, 35J40

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