We characterize regular boundary points for p-harmonic functions using weak barriers. We use this to obtain some consequences on boundary regularity. The results also
hold for -harmonic functions under the usual assumptions on , and for Cheeger p-harmonic functions in metric spaces.
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Let R be a local ring and let (x1, …, xr) be part of a system of parameters of a finitely generated R-module M, where r < dimRM. We will show that if (y1, …, yr) is part of a reducing system of parameters of M with (y1, …, yr) M = (x1, …, xr) M then (x1, …, xr) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ VR(x1, …, xr) with dimRR/P = dimRM − r the localization MP of M at P is an r-dimensional Cohen-Macaulay module over RP.
Furthermore, we will show that M is a Cohen-Macaulay module iff yd is a non zero divisor on M/(y1, …, yd−1) M, where (y1, …, yd) is a reducing system of parameters of M (d:= dimRM). 相似文献
For Ξ∈Rn,t∈R andf∈S(Rn) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys0 such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(Rn) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|2) is replaced by exp(it|ξ|a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL2(Rn) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity. 相似文献
We prove that any polyhedron in two dimensions admits a type of potential theoretic skeleton called mother body. We also show that the mother bodies of any polyhedron in any number of dimensions are in one-to-one correspondence with certain kinds of decompositions of the polyhedron into convex subpolyhedra. A consequence of this is that there can exist at most finitely many mother bodies of any given polyhedron. The main ingredient in the proof of the first mentioned result consists of showing that any polyhedron in two dimensions contains a convex subpolyhedron which sticks to it in the sense that every face of the subpolyhedron has some part in common with a face of the original polyhedron. 相似文献
An experimental and numerical investigation of a confined laminar inverse diffusion flame (IDF) with pure oxygen as oxidizer and carbon dioxide diluted methane as fuel with a global stoichiometry of partial oxidation processes (equivalence ratio of 2.5) is presented. The present burner setup allows studying both the flame and the post-flame zone in a simplified geometry considering typical operating conditions as found in large-scale gasifiers. This partial oxidation flame setup is characterized by very high temperatures close to the stoichiometric oxidation zone due to oxy-fuel combustion, whereas lower temperatures and slow endothermic post-flame conversion reactions with long residence times are found in the fuel rich post-flame region. The scope of this paper is to investigate different modeling approaches suitable for both regimes by comparing the simulation results to detailed experimental data. Planar OH laser-induced fluorescence (OH-LIF) was performed for measuring the hydroxyl radical in the reaction zone and the results are compared to CFD calculations. Based on this comparison, the necessary level of detail of diffusion flux modeling, which includes Soret and Dufour effects, is analyzed and established. Finally, steady and unsteady non-premixed flamelet approaches based on a single mixture fraction are used in order to study their applicability for both the oxidation and post-flame zone. Significantly different time scales are obtained using different flamelet paths. Their influence on the results is investigated in the steady flamelet and the Lagrangian flamelet approach. 相似文献
Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results. 相似文献
For optimal control problems subject to index-one differential-algebraic equations in semi-explicit form we discuss second order sufficient conditions in form of a coercivity condition taking into account the two-norm discrepancy. Furthermore we introduce a related Riccati-type and Legendre-Clebsch condition which are sufficient for the validity of the coercivity condition. Using the implicit Euler-discretization we approximate the optimal control problem and analyze the convergence of solutions of the local minimum principle for the discretized optimal control problem by applying the general convergence framework of Stetter, which requires the discretization method to be continuous, consistent, and stable.