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2020
The exterior Dirichlet problem consists in finding u such that $$\displaystyle \begin {array}{rl} &Au(x)=0,\quad x\in S^-,\\ &u(x)=\mathscr {D}(x),\quad x\in {\partial S},\\ &u\in \mathscr {A}^*, \end {array} $$ where the vector function \(\mathscr {D}\in C^{(1,\alpha )}({\partial S})\), α ∈ (0, 1), is prescribed.
Christian Constanda, Dale Doty
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The exterior Dirichlet problem consists in finding u such that $$\displaystyle \begin {array}{rl} &Au(x)=0,\quad x\in S^-,\\ &u(x)=\mathscr {D}(x),\quad x\in {\partial S},\\ &u\in \mathscr {A}^*, \end {array} $$ where the vector function \(\mathscr {D}\in C^{(1,\alpha )}({\partial S})\), α ∈ (0, 1), is prescribed.
Christian Constanda, Dale Doty
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2020
The interior Dirichlet problem consists of the equation and boundary condition $$\displaystyle \begin {array}{ll} &Au(x)=0,\quad x\in S^+,\\ &u(x)=\mathscr {D}(x),\quad x\in {\partial S}, \end {array} $$ where the vector function \(\mathscr {D}\in C^{1,\alpha }({\partial S})\), α ∈ (0, 1), is prescribed.
Christian Constanda, Dale Doty
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The interior Dirichlet problem consists of the equation and boundary condition $$\displaystyle \begin {array}{ll} &Au(x)=0,\quad x\in S^+,\\ &u(x)=\mathscr {D}(x),\quad x\in {\partial S}, \end {array} $$ where the vector function \(\mathscr {D}\in C^{1,\alpha }({\partial S})\), α ∈ (0, 1), is prescribed.
Christian Constanda, Dale Doty
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2001
In its simplest form the Dirichlet problem may be stated as follows: for a given function \( f \in C\left( {{\partial ^\infty }\Omega } \right)\), determine, if possible, a function h ∈ H(Ω) such that h(x) → f(y) as x → y for each \( y \in {\partial ^\infty }\Omega \).
David H. Armitage, Stephen J. Gardiner
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In its simplest form the Dirichlet problem may be stated as follows: for a given function \( f \in C\left( {{\partial ^\infty }\Omega } \right)\), determine, if possible, a function h ∈ H(Ω) such that h(x) → f(y) as x → y for each \( y \in {\partial ^\infty }\Omega \).
David H. Armitage, Stephen J. Gardiner
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2014
The natural Dirichlet problem in C n is not the classical one for solutions of the Laplace equation but rather for solutions of the Monge–Ampere equation. We present the solution of this Dirichlet problem in a ball.
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The natural Dirichlet problem in C n is not the classical one for solutions of the Laplace equation but rather for solutions of the Monge–Ampere equation. We present the solution of this Dirichlet problem in a ball.
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Dirichlet problem for demi-coercive functionals
Nonlinear Analysis: Theory, Methods & Applications, 1986A class of variational integrals with integrands satisfying a linear growth condition is called by the authors demi-coercive. Example of demi- coercive functionals arise naturally for instance in studying the equilibrium of elastic structures with unilateral constraints on the stress, and in studying the nonparametric Plateau problem.
ANZELLOTTI G +2 more
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An area‐Dirichlet integral minimization problem
Communications on Pure and Applied Mathematics, 2001zbMATH Open Web Interface contents unavailable due to conflicting licenses.
I. Athanasopoulos +3 more
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Theoretical and Mathematical Physics
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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1970
Let d(n) denote the number of positive divisors of the positive integer n. Let $$E(x) = \sum\limits_{n \leqslant x} {d(n) - x\log x - (2\gamma - 1)x,\,x \geqslant 1}$$ where γ is Euler’s constant. It is known, after Dirichlet, that $$ E(x) = 0({x^{{\frac{1}{2}}}}),\quad as\quad x \to \infty $$
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Let d(n) denote the number of positive divisors of the positive integer n. Let $$E(x) = \sum\limits_{n \leqslant x} {d(n) - x\log x - (2\gamma - 1)x,\,x \geqslant 1}$$ where γ is Euler’s constant. It is known, after Dirichlet, that $$ E(x) = 0({x^{{\frac{1}{2}}}}),\quad as\quad x \to \infty $$
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2009
In this chapter, approximation of solutions of Laplace’s equation requires the study of sequences of harmonic functions, making use of the integral representations and averaging properties of harmonic functions. The latter property is used to incorporate a larger class of functions called superharmonic functions that are used to approximate solutions ...
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In this chapter, approximation of solutions of Laplace’s equation requires the study of sequences of harmonic functions, making use of the integral representations and averaging properties of harmonic functions. The latter property is used to incorporate a larger class of functions called superharmonic functions that are used to approximate solutions ...
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Dirichlet's problem in banach space
Mathematical Notes of the Academy of Sciences of the USSR, 1983The author investigates a Dirichlet problem for the equation \(Lu=tr A(x)u''(x)A(x)+u'(x)a(x)=-g(x)\) in a region G of a Banach space X with boundary \(\Gamma\). Using a known probability representation of the solution of the problem \(Lu=-g\), \(u|_{\Gamma}=\psi\) he extends some results by \textit{N. N. Frolov} [Teor. Veroyatn. Mat. Stat. 3, 200-210 (
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