The Dirichlet problem for the complex Hessian operator in the class $\mathcal{N}_m(\Omega,f)$

2021 ◽  
Vol 127 (2) ◽  
pp. 287-316
Author(s):  
Ayoub El Gasmi
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Borel Measure ◽  
Positive Borel Measure ◽  
Hessian Equation ◽  
Hessian Operator ◽  
Hyperconvex Domain ◽  
Complex Hessian Operator

Let $\Omega\subset \mathbb{C}^{n}$ be a bounded $m$-hyperconvex domain, where $m$ is an integer such that $1\leq m\leq n$. Let $\mu$ be a positive Borel measure on $\Omega$. We show that if the complex Hessian equation $H_m (u) = \mu$ admits a (weak) subsolution in $\Omega$, then it admits a (weak) solution with a prescribed least maximal $m$-subharmonic majorant in $\Omega$.

scholarly journals Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

2021 ◽  
Vol 15 (3) ◽  
Author(s):  
Changbao Pang ◽  
Antti Perälä ◽  
Maofa Wang
Keyword(s):  
Borel Measure ◽  
Bergman Spaces ◽  
Embedding Theorem ◽  
Weighted Bergman Spaces ◽  
Embedding Theorems ◽  
Positive Borel Measure ◽  
Doubling Property ◽  
Area Operator ◽  
Special Cases ◽  
Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

Embeddings of Müntz Spaces in

2019 ◽  
Vol 62 (1) ◽  
pp. 1-9
Author(s):  
Ihab Al Alam ◽  
Pascal Lefèvre
Keyword(s):  
Borel Measure ◽  
Essential Norm ◽  
Weak Compactness ◽  
Positive Borel Measure ◽  
Absolutely Continuous ◽  
Borel Measures ◽  
Embedding Operator ◽  
Müntz Spaces

AbstractIn this paper, we discuss the properties of the embedding operator $i_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6EC}}:M_{\unicode[STIX]{x1D6EC}}^{\infty }{\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707})$, where $\unicode[STIX]{x1D707}$ is a positive Borel measure on $[0,1]$ and $M_{\unicode[STIX]{x1D6EC}}^{\infty }$ is a Müntz space. In particular, we compute the essential norm of this embedding. As a consequence, we recover some results of the first author. We also study the compactness (resp. weak compactness) and compute the essential norm (resp. generalized essential norm) of the embedding $i_{\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2}}:L^{\infty }(\unicode[STIX]{x1D707}_{1}){\hookrightarrow}L^{\infty }(\unicode[STIX]{x1D707}_{2})$, where $\unicode[STIX]{x1D707}_{1}$, $\unicode[STIX]{x1D707}_{2}$ are two positive Borel measures on [0, 1] with $\unicode[STIX]{x1D707}_{2}$ absolutely continuous with respect to $\unicode[STIX]{x1D707}_{1}$.

scholarly journals Flows of measures generated by vector fields

2018 ◽  
Vol 148 (4) ◽  
pp. 773-818
Author(s):  
Emanuele Paolini ◽  
Eugene Stepanov
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Borel Measure ◽  
Weak Sense ◽  
Positive Borel Measure ◽  
Smooth Structure ◽  
Smooth Vector ◽  
Integral Curves ◽  
The Given ◽  
Measurable Vector

The scope of the paper is twofold. We show that for a large class of measurable vector fields in the sense of Weaver (i.e. derivations over the algebra of Lipschitz functions), called in the paper laminated, the notion of integral curves may be naturally defined and characterized (when appropriate) by an ordinary differential equation. We further show that for such vector fields the notion of a flow of the given positive Borel measure similar to the classical one generated by a smooth vector field (in a space with smooth structure) may be defined in a reasonable way, so that the measure ‘flows along’ the appropriately understood integral curves of the given vector field and the classical continuity equation is satisfied in the weak sense.

scholarly journals Nonstandard Dirichlet problems with competing (p,q)-Laplacian, convection, and convolution

2021 ◽  
Vol 66 (1) ◽  
pp. 95-103
Author(s):  
Dumitru Motreanu ◽  
Viorica Venera Motreanu
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Integrable Function ◽  
Generalized Solution ◽  
Dirichlet Problems ◽  
Reaction Term ◽  
Fixed Point Approach

"The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integrable function. We prove the existence of a generalized solution through a combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak solution to the respective elliptic problem."

scholarly journals On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Nguyen Huu Nhan ◽  
Le Thi Phuong Ngoc ◽  
Nguyen Thanh Long
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Small Parameter ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Existence And Uniqueness ◽  
Linearization Method ◽  
Asymptotic Expansion Of Solution

We consider the Robin-Dirichlet problem for a nonlinear wave equation of Kirchhoff-Carrier type. Using the Faedo-Galerkin method and the linearization method for nonlinear terms, the existence and uniqueness of a weak solution are proved. An asymptotic expansion of high order in a small parameter of a weak solution is also discussed.

scholarly journals Some dual aspects of the Poisson kernel

1990 ◽  
Vol 33 (2) ◽  
pp. 207-232 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Continuous Function ◽  
Borel Measure ◽  
Poisson Kernel ◽  
Infinite Subset ◽  
Open Unit ◽  
Positive Borel Measure ◽  
Positive Continuous Function ◽  
Open Unit Disc ◽  
Counting Measure ◽  
Countably Infinite

The Poisson kernel is defined for z in the open unit disc D and ζ in the unit circle ∂D. As usually employed, it is integrated with respect to the second variable and a measure on ∂D to yield a harmonic function on D. Here, we fix a σ-finite positive Borel measure m on D and integrate the Poisson kernel with respect to the first variable against a function φ in L1(m) to obtain a function Tmφ on ∂D. We ask for what measures m the range of Tm is L1(∂D), for what m the kernel of Tm is non-zero, and for what m every positive continuous function on ∂D is of the form Tmφ with φ non-negative. When m is the counting measure of a countably infinite subset {ak:k∈ℕ} of D, the function (Tmφ)(ζ) is of the form with . The main results generalize results previously obtained for sums of this form. A related mapping from Lp(m) into Lp(∂D) with 1 <p<∞ is briefly considered.

scholarly journals The 2D Dirichlet Problem for the Propagative Helmholtz Equation in an Exterior Domain with Cracks and Singularities at the Edges

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
P. A. Krutitskii
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Integral Representation ◽  
Helmholtz Equation ◽  
Exterior Domain ◽  
Index Zero ◽  
Compatibility Conditions ◽  
Unique Classical Solution ◽  
Asymptotic Formulae ◽  
Solution Gradient

The Dirichlet problem for the 2D Helmholtz equation in an exterior domain with cracks is studied. The compatibility conditions at the tips of the cracks are assumed. The existence of a unique classical solution is proved by potential theory. The integral representation for a solution in the form of potentials is obtained. The problem is reduced to the Fredholm equation of the second kind and of index zero, which is uniquely solvable. The asymptotic formulae describing singularities of a solution gradient at the edges (endpoints) of the cracks are presented. The weak solution to the problem may not exist, since the problem is studied under such conditions that do not ensure existence of a weak solution.

The Range of the Cesàro Operator Acting on

2019 ◽  
Vol 63 (3) ◽  
pp. 633-642
Author(s):  
Guanlong Bao ◽  
Hasi Wulan ◽  
Fangqin Ye
Keyword(s):  
Unit Disk ◽  
Borel Measure ◽  
Dirichlet Space ◽  
Invariant Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
Positive Borel Measure ◽  
Cesàro Operator ◽  
Borel Measures ◽  
Cesaro Operator

AbstractIn 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on $H^{\infty }$; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$, but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$. In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$. In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures $\unicode[STIX]{x1D707}_{0}$. Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.

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