Some Analytic and Geometric Properties of Solution to Skew-Symmetric Elliptic Systems

2021 ◽  
pp. 4-17
Author(s):  
A.O. Bagapsh
Keyword(s):  
Elliptic System ◽  
Complex Variable ◽  
Elliptic Systems ◽  
Second Order ◽  
Singular Points ◽  
Laurent Expansion ◽  
Geometric Properties ◽  
Complex Valued ◽  
Strongly Elliptic System

We study the properties of complex-valued functions of a complex variable, whose real and imaginary parts satisfy a second-order skew-symmetric strongly elliptic system with constant real coefficients in the plane. The behavior of such functions and their dilatations near singular points is investigated and the dependence of the type of the singularity on the form of the Laurent expansion of the function under consideration is established. The principle of the argument is established for the functions with poles under study, analogs of the Ruschet and Hurwitz theorems are proved

scholarly journals Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

2021 ◽  
Author(s):  
Gioconda Moscariello ◽  
Giulio Pascale
Keyword(s):  
Dirichlet Problem ◽  
Unit Ball ◽  
Elliptic System ◽  
Positive Constant ◽  
Elliptic Systems ◽  
Second Order ◽  
Identity Matrix ◽  
Bmo Coefficients

AbstractWe consider linear elliptic systems whose prototype is $$\begin{aligned} div \, \Lambda \left[ \,\exp (-|x|) - \log |x|\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$ d i v Λ exp ( - | x | ) - log | x | I D u = d i v F + g in B . Here B denotes the unit ball of $$\mathbb {R}^n$$ R n , for $$n > 2$$ n > 2 , centered in the origin, I is the identity matrix, F is a matrix in $$W^{1, 2}(B, \mathbb {R}^{n \times n})$$ W 1 , 2 ( B , R n × n ) , g is a vector in $$L^2(B, \mathbb {R}^n)$$ L 2 ( B , R n ) and $$\Lambda $$ Λ is a positive constant. Our result reads that the gradient of the solution $$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$ u ∈ W 0 1 , 2 ( B , R n ) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant $$\Lambda $$ Λ is not large enough.

scholarly journals Some results of Fekete–Szegö type for Bavrin’s families of holomorphic functions in $${\mathbb {C}}^{n}$$

2021 ◽  
Author(s):  
Renata Długosz ◽  
Piotr Liczberski
Keyword(s):  
Complex Variable ◽  
Univalent Functions ◽  
Holomorphic Functions ◽  
Symmetry Condition ◽  
Type Problem ◽  
Geometric Properties ◽  
Several Variables ◽  
Starlike Mappings ◽  
Complex Valued

AbstractIn the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a (j, k)-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in $${\mathbb {C}}^{n}.$$ C n .

scholarly journals Regularity of the extremal solutions associated with elliptic systems

2018 ◽  
Vol 149 (04) ◽  
pp. 1037-1046
Author(s):  
A. Aghajani ◽  
C. Cowan
Keyword(s):  
Convex Function ◽  
Elliptic System ◽  
Bounded Domain ◽  
Elliptic Systems ◽  
Extremal Solution ◽  
Second Order ◽  
Extremal Solutions ◽  
Smooth Bounded Domain ◽  
Order Polynomial ◽  
Image Position

AbstractWe examine the elliptic system given by$$\left\{ {\matrix{ {-\Delta u = \lambda f(v)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {-\Delta v = \gamma f(u)} \hfill &amp; {{\rm in }\,\,\Omega ,} \hfill \cr {u = v = 0} \hfill &amp; {{\rm on }\,\,\partial \Omega ,} \hfill \cr } } \right.$$where λ, γ are positive parameters, Ω is a smooth bounded domain in ℝNandfis aC2positive, nondecreasing and convex function in [0, ∞) such thatf(t)/t→ ∞ ast→ ∞. Assuming$$0 < \tau _-: = \mathop {\lim \inf }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les \tau _ + : = \mathop {\lim \sup }\limits_{t\to \infty } \displaystyle{{f(t){f}^{\prime \prime}(t)} \over {{f}^{\prime}{(t)}^2}} \les 2,$$we show that the extremal solution (u*,v*) associated with the above system is smooth provided thatN&lt; (2α*(2 − τ+) + 2τ+)/(τ+)max{1, τ+}, where α*&gt; 1 denotes the largest root of the second-order polynomial$$[P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).]$$As a consequence,u*,v* ∈L∞(Ω) forN&lt; 5. Moreover, if τ−= τ+, thenu*,v* ∈L∞(Ω) forN&lt; 10.

Positivity results for a nonlocal elliptic equation

1998 ◽  
Vol 128 (4) ◽  
pp. 697-715 ◽  
Author(s):  
Pedro Freitas ◽  
Guido Sweers
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Parabolic Equations ◽  
Elliptic Systems ◽  
Stationary Solutions ◽  
Second Order ◽  
Real Eigenvalue ◽  
Nonlocal Operator ◽  
Nonlocal Parabolic Equations ◽  
Positive Eigenfunction

In this paper we consider a second-order linear nonlocal elliptic operator on a bounded domain in ℝn (n ≧ 3), and give conditions which ensure that this operator has a positive inverse. This generalises results of Allegretto and Barabanova, where the kernel of the nonlocal operator was taken to be separable. In particular, our results apply to the case where this kernel is the Green's function associated with second-order uniformly elliptic operators, and thus include the case of some linear elliptic systems. We give several other examples. For a specific case which appears when studying the linearisation of nonlocal parabolic equations around stationary solutions, we also consider the associated eigenvalue problem and give conditions which ensure the existence of a positive eigenfunction associated with the smallest real eigenvalue.

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