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Register a new userSome properties of a class of elliptic partial differential operators
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We prove Schwarz-Pick type estimates and coefficient estimates for a class of elliptic partial differential operators introduced by Olofsson. Then we apply these results to obtain a Landau type theorem.
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The concept of moment differentiation is extended to the class of moment summable functions, giving rise to moment differential properties. The main result leans on accurate upper estimates for the integral representation of the moment derivatives of functions under exponential-like growth at infinity, and appropriate deformation of the integration paths. The theory is applied to obtain summability results of certain family of generalized linear moment partial differential equations with variable coefficients.
Let $f$ be a transcendental meromorphic function defined in the complex plane $mathbb{C}$. We consider the value distribution of the differential polynomial $f^{q_{0}}(f^{(k)})^{q_{k}}$, where $q_{0}(geq 2), q_{k}(geq 1)$ are $k(geq1)$ non-negative integers. We obtain a quantitative estimation of the characteristic function $T(r, f)$ in terms of $overline{N}left(r,frac{1}{f^{q_{_{0}}}(f^{(k)})^{q_{k}}-1}right)$.par Our result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., 14, 93-100, 2011) and Karmakar and Sahoo (Results Math., 73, 2018) for a particular class of transcendental meromorphic functions.
For a constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schrodinger operator as well as non-degenerate parabolic differential operators like the heat operator we characterize when open subsets $X_1subseteq X_2$ of $mathbb{R}^d$ form a $P$-Runge pair. The presented condition does not require any kind of regularity of the boundaries of $X_1$ nor $X_2$. As part of our result we prove that for a large class of non-elliptic operators $P(D)$ there are smooth solutions $u$ to the equation $P(D)u=0$ on $mathbb{R}^d$ with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for $P(D)$.
Let s 1 ,. .. , s k be the elementary symmetric functions of the complex variables x 1 ,. .. , x k. We say that F $in$ C[s 1 ,. .. , s k ] is a trace function if their exists f $in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f (x j) for all s $in$ C k. We give an explicit finite family of second order differential operators in the Weyl algebra W 2 := C[s 1 ,. .. , s k ] $partial$ $partial$s 1 ,. .. , $partial$ $partial$s k which generates the left ideal in W 2 of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer k a holonomic system which is a quotient of W 2 by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: z k + k h=1 (--1) h .s h .z k--h = 0.
This paper presents an observation that under reasonable conditions, many partial differential equations from mathematical physics possess three structural properties. One of them can be understand as a variant of the celebrated Onsager reciprocal relation in Modern Thermodynamics. It displays a direct relation of irreversible processes to the entropy change. We show that the properties imply various entropy dissipation conditions for hyperbolic relaxation problems. As an application of the observation, we propose an approximation method to solve relaxation problems. Moreover, the observation is interpreted physically and verified with eight (sets of) systems from different fields.
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