Potential Theoretic Methods Research Articles

Zinc Oxide–Zinc Phthalocyanine Interface for Hybrid Solar Cells

The structural, electronic, and optical properties of a hybrid interface formed by zinc phthalocyanine (ZnPc) molecules adsorbed on the (1010) zinc oxide (ZnO) surface have been investigated by using ab initio and model potential theoretical methods. In particular, the attention has been focused on the effects of molecular assembling on the interface properties by considering cofacial and planar molecular aggregates on the surface. Present results show that planar aggregations provide a remarkable molecule-to-surface electronic coupling which can favor electron injection toward the substrate. Furthermore, we predict a blue shift of absorption bands in the case of cofacial aggregation and a red shift in the case of nanostructured planar J-stripes, which are in agreement with previous phenomenological models and give a firm theoretical support to observed relationships between red shift and molecular assembling. All together, present results indicate that structural and electronic properties can be achieve...

Positive solutions of Schrödinger equations and fine regularity of boundary points

Given a Lipschitz domain $\Omega $ in ${\mathbb R} ^N $ and a nonnegative potential $V$ in $\Omega $ such that $V(x)\, d(x,\partial \Omega)^2$ is bounded in $\Omega $ we study the fine regularity of boundary points with respect to the Schr\"odinger operator $L_V:= \Delta -V$ in $\Omega $. Using potential theoretic methods, several conditions equivalent to the fine regularity of $z \in \partial \Omega $ are established. The main result is a simple (explicit if $\Omega $ is smooth) necessary and sufficient condition involving the size of $V$ for $z$ to be finely regular. An essential intermediate result consists in a majorization of $\int_A | {\frac {u} {d(.,\partial \Omega)}} | ^2\, dx$ for $u$ positive harmonic in $\Omega $ and $A \subset \Omega $. Conditions for almost everywhere regularity in a subset $A $ of $ \partial \Omega $ are also given as well as an extension of the main results to a notion of fine ${\mathcal L}_1 | {\mathcal L}_0$-regularity, if ${\mathcal L}_j={\mathcal L}-V_j$, $V_0,\, V_1$ being two potentials, with $V_0 \leq V_1$ and ${\mathcal L}$ a second order elliptic operator.

Potential-theoretical methods in the construction of measure-valued Markov branching processes

We develop potential-theoretical methods in the construction of measure-valued branching processes.We complete results of P. J. Fitzsimmons and E. B. Dynkin on the construction, regularity and other properties of the superprocess associated with a given right process and a branching mechanism.

Distribution of algebraic numbers

Schur studied limits of the arithmetic means A n of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk. If the leading coefficients are bounded, Schur proved that . We show that A n → 0, and estimate the rate of convergence by generalizing the Erdős–Turán theorem on the distribution of zeros. As an application, we show that integer polynomials have some unexpected restrictions of growth on the unit disk. Schur also studied problems on means of algebraic numbers on the real line. When all conjugate algebraic numbers are positive, the problem of finding the sharp lower bound for was developed further by Siegel and others. We provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line. Potential theoretic methods allow us to consider the distribution of algebraic numbers in or near general sets in the complex plane. We introduce the generalized Mahler measure, and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one. The quantitative aspects of this equidistribution are also analyzed in terms of the generalized Mahler measure.

Landau's Theorem for Holomorphic Curves in Projective Space and The Kobayashi Metric on Hyperplane Complements

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in n-dimensional complex projective space, we find an explicit constant K such that for every holomorphic map f from the unit disc to the complement of these hyperplanes, the derivative of f at the origin measured with respect to the Fubuni-Study metric is bouned above by K. This result gives an explicit lower bound on the Royden function, i.e., the ratio of the Kobayashi metric on the hyperplane complement to the Fubini-Study metric. Our estimate is based on the potential-theoretic method of Eremenko and Sodin.

The graph and range singularity spectra of random wavelet series built from Gibbs measures

We consider the random wavelet series built from Gibbs measures, and study the Hausdorff dimension of the graph and range of these functions restricted to their iso-Hölder sets. To obtain the Hausdorff dimension of these sets, we apply the potential theoretic method to families of Gibbs measures defined on a sequence of topologically transitive subshift of finite type whose Hausdorff distance to the set of zeros of the mother wavelet tends to 0.

Homogeneous nucleation for Glauber and Kawasaki dynamics in large volumes at low temperatures

In this paper, we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let β denote the inverse temperature and let Λ β C ℤ 2 be a square box with periodic boundary conditions such that lim β→∞ |Λ β | = ∞. We run the dynamics on Λ β , starting from a random initial configuration where all of the droplets (clusters of plus-spins and clusters of particles, respectively) are small. For large β and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single critical droplet somewhere in Λ β . Using potential-theoretic methods, we compute the average nucleation time (the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to 1 as β → ∞. It turns out that this time grows as Ke Γβ /|Λ β | for Glauber dynamics and as Kβe Γβ /|Λ β | for Kawasaki dynamics, where Γ is the local canonical (resp. grand-canonical) energy, to create a critical droplet and K is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on |Λ β |). The fact that the average nucleation time is inversely proportional to |Λ β | is referred to as homogeneous nucleation because it says that the critical droplet for the transition appears essentially independently in small boxes that partition Λ β .

Analytical Model for Comb-Capacitance Fringe Fields

Analytical expressions for electric potential and electric fringe fields in regions above the fingers of MEMS (microelectromechanical systems) comb capacitances are derived using potential-theoretic methods. The formulas are valid for the following: (1) a comb geometry exhibiting a large number of identical fingers and (2) a finger geometry where the gap between fingers is small compared to the height of the fingers and the finger overlap. For these conditions, symmetries that are inherent to the comb geometry can be exploited fruitfully to set up a properly defined Dirichlet problem formulation for the potential which can be solved for explicitly, yielding a series expansion for the electrostatic potential and electric field components. The accuracy of the approximated analytical solutions, obtained by truncating the series expansions to contain only a finite number of terms, is compared with the results obtained from finite element simulations of the electrostatic potential and electric field. From the analytic result, an approximation to the levitation force acting on the upper finger surfaces is derived. A formula expressing the mean length of the fringe electric field lines emanating from the upper finger surfaces into the ambient space is presented.

Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis)

Haïm Brezis and Augusto Ponce introduced and studied several extensions of Kato's inequality, in particular Kato's inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a W1,1 function in a smooth domain [H. Brezis, A.C. Ponce, Kato's inequality when Δu is a measure, C. R. Acad. Sci. Paris Sér. I 338 (2004) 599–604; H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241]. Using potential theoretic methods we answer here some questions raised in [H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217–1241] about the relations between the normal derivative of a function u and the normal derivative of its positive part u+. The results apply to a large class of domains and elliptic operators in divergence form and finally an expression of the normal derivative of a function of u is given. In the final appendix, H. Brezis solves an old question of J. Serrin about pathological solutions of certain elliptic equations [J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3) 18 (1964) 385–387]. This is used in the paper to extend the first version of our main result.

Brownian couplings, convexity, and shy-ness

Benjamini, Burdzy and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least a given positive distance away from each other for all time. Among other results, they showed that no shy couplings could exist for reflected Brownian motions in $C^2$ bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al.(2007) result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary.

Feynman-Kac Formula for Left Continuous Additive Functionals and Extended Kato Class Measures

The perturbation of the generator of a Borel right process by a signed measure is investigated, using probabilistic and analytic potential theoretical methods. We establish a Feynman-Kac formula associated with measures charging no polar set and belonging to an extended Kato class. A main tool of this approach is the validity of a Khas’minskii Lemma for Stieltjes exponentials of positive left continuous additive functionals.

The Uniqueness Problem for Subordinate Resolvents with Potential Theoretical Methods

We present an analytic version of the following uniqueness problem for Markov processes: if two subprocesses of a given transient, Borel right process have the same excessive functions then they coincide. Our treatment is given in terms of subordinate sub-Markovian resolvents of kernels in the sense of P. A. Mayer and uses essentially the subordination operators introduced by G. Mokobodzki.

Inégalité de Kato et inégalité de Kato jusqu'au bord

Haïm Brezis and Augusto Ponce introduced and studied in their works several extensions of Kato's inequality, in particular Kato's inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a function. Using Potential theoretic methods we answer here some questions raised in Brezis and Ponce (2008). To cite this article: A. Ancona, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The Hausdorff dimension of pulse-sum graphs

AbstractWe consider random functions formed as sums of pulses where Xn are independent random vectors, 0<α<1, and G is an elementary “pulse” or “bump”. Typically such functions have fractal graphs and we find the Hausdorff dimension of these graphs using a novel variant on the potential theoretic method.

A systematic derivation of Green's and Poisson's matrices for boundary-value problems in elastostatic half-spaces

Green's and Poisson's matrices in an n-dimensional half-space are computed for the system of linear isotropic elasticity and four types of boundary conditions attacked by Boussinesq and Cerruti by means of potential-theoretic methods and reconsidered by Sneddon (1993). This involves the solution of convolution equations and the computation of various convolution products by means of distribution theory. The question of uniqueness is also addressed.

On the structure of partial balayage

We define partial balayage, construct it with classical potential-theoretic methods and establish the so called structure formula in full generality. Finally we give a stochastic construction of partial balayage.

Transform Potential-Theoretic Method for Acoustic Radiation from Structures

This paper presents further development in a new time-domain computational technique developed for the determination of far-field acoustic radiation. This method, referred to as the transform potential-theoretic (TPT) method, solves linear time-dependent wave propagation problems in an unbounded medium, combining techniques involving numerical transforms/inverse transforms and potential theory. The end result is a robust procedure that is accurate and computationally efficient. The TPT technique is meshless, yet it can handle arbitrary geometries for the acoustic source. The validity of the procedure is demonstrated for the case of an acoustic wave propagation from a two-dimensional bounded surface embedded in a uniform flow. Both direct simulations as well as the current method(s) are compared. The single-layer potential theory based TPT method required fewer points on the boundary than the double-layer potential theory based TPT and performs better than direct simulation in high subsonic Mach number flows (M=0.8). The times required to compute these solutions were more than two orders of magnitude lower than direct simulation.

Potential theoretic methods for the hyperbolic Maxwell system in rough cylinders

AbstractThis paper is concerned with the solvability of the hyperbolic Maxwell system in Lipschitz cylinders via integral methods. By means of the Laplace transform, the original hyperbolic Maxwell system is reduced to a family of stationary Maxwell equations indexed by ω, the Laplace variable. The crux of the matter is establishing how all relevant constants in the main estimates for this family of Maxwell equations depend on ω. Among other things, appropriate norms and weighted Sobolev spaces are considered so that the integral equation method works. The solution is expressed in terms of retarded potentials. Copyright © 2004 John Wiley & Sons, Ltd.

Nodal domain theorems à la Courant

Let H(\Omega_0)=-\Delta+V be a Schrödinger operator on a bounded domain \Omega_0\subset \mathbb R^d (d\geq 2) with Dirichlet boundary condition. Suppose that \Omega_\ell (\ell \in \{1,\dots,k\}) are some pairwise disjoint subsets of \Omega_0 and that H(\Omega_\ell) are the corresponding Schrödinger operators again with Dirichlet boundary condition. We investigate the relations between the spectrum of H(\Omega_0) and the spectra of the H(\Omega_\ell) . In particular, we derive some inequalities for the associated spectral counting functions which can be interpreted as generalizations of Courant's nodal theorem. For the case where equality is achieved we prove converse results. In particular, we use potential theoretic methods to relate the \Omega_\ell to the nodal domains of some eigenfunction of H(\Omega_0) .

An inequality for the norm of a polynomial factor

Let p ( z ) p(z) be a monic polynomial of degree n n , with complex coefficients, and let q ( z ) q(z) be its monic factor. We prove an asymptotically sharp inequality of the form ‖ q ‖ E ≤ C n ‖ p ‖ E \|q\|_{E} \le C^n \, \|p\|_E , where ‖ ⋅ ‖ E \|\cdot \|_E denotes the sup norm on a compact set E E in the plane. The best constant C E C_E in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.