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1. CJM 2013 (vol 66 pp. 429)

Rivera-Noriega, Jorge
 Perturbation and Solvability of Initial $L^p$ Dirichlet Problems for Parabolic Equations over Non-cylindrical Domains For parabolic linear operators $L$ of second order in divergence form, we prove that the solvability of initial $L^p$ Dirichlet problems for the whole range $1\lt p\lt \infty$ is preserved under appropriate small perturbations of the coefficients of the operators involved. We also prove that if the coefficients of $L$ satisfy a suitable controlled oscillation in the form of Carleson measure conditions, then for certain values of $p\gt 1$, the initial $L^p$ Dirichlet problem associated to $Lu=0$ over non-cylindrical domains is solvable. The results are adequate adaptations of the corresponding results for elliptic equations. Keywords:initial $L^p$ Dirichlet problem, second order parabolic equations in divergence form, non-cylindrical domains, reverse HÃ¶lder inequalitiesCategory:35K20

2. 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 SchrÃ¶dinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in HÃ¶lder spaces. Keywords:standing waves, elliptic regularity, Lipschitz domainCategories:35J25, 35J65

3. CJM 2006 (vol 58 pp. 492)

Chua, Seng-Kee
 Extension Theorems on Weighted Sobolev Spaces and Some Applications We extend the extension theorems to weighted Sobolev spaces $L^p_{w,k}(\mathcal D)$ on $(\varepsilon,\delta)$ domains with doubling weight $w$ that satisfies a Poincar\'e inequality and such that $w^{-1/p}$ is locally $L^{p'}$. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities. Keywords:PoincarÃ© inequalities, $A_p$ weights, doubling weights, $(\ep,\delta)$ domain, $(\ep,\infty)$ domainCategory:46E35

4. CJM 2006 (vol 58 pp. 401)

Kolountzakis, Mihail N.; Révész, Szilárd Gy.
 On Pointwise Estimates of Positive Definite Functions With Given Support The following problem has been suggested by Paul Tur\' an. Let $\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$ or in the torus $\TT^d$. Then, what is the largest possible value of the integral of positive definite functions that are supported in $\Omega$ and normalized with the value $1$ at the origin? From this, Arestov, Berdysheva and Berens arrived at the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, the supremum of possible function values at $z$ is to be found for any given point $z\in\Omega$. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\RR^d$ and to non-convex domains as well. Here we present another approach to the problem, giving the solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we elaborate on the fact that the problem is essentially one-dimensional and investigate non-convex open domains as well. We show that the extremal problems are equivalent to some more familiar ones concerning trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relationship between the problem for $\RR^d$ and that for $\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hierarchy of difficulty is established, so that extremal problems for trigonometric polynomials gain renewed recognition. Keywords:Fourier transform, positive definite functions and measures, TurÃ¡n's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problemsCategories:42B10, 26D15, 42A82, 42A05

5. CJM 2004 (vol 56 pp. 655)

Tao, Xiangxing; Wang, Henggeng
 On the Neumann Problem for the SchrÃ¶dinger 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 $1-\epsilon Keywords:Neumann problem, SchrÃ¶dinger equation, Lipschitz, domain, reverse HÃ¶lder class,$H^p$spaceCategories:42B20, 35J10 6. CJM 2002 (vol 54 pp. 1121) Bao, Jiguang  Fully Nonlinear Elliptic Equations on General Domains By means of the Pucci operator, we construct a function$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 lemmaCategories:35D05, 35D10, 35J60, 35J67
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