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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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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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On a Constrained Dirichlet Problem
SIAM Journal on Control and Optimization, 2002Summary: We consider a Dirichlet minimum problem with a pointwise constraint on the gradient, i.e., \(\| \nabla u(x)\| \leq 1\) a.e., or, equivalently, an unconstrained minimum problem with an extended-valued integrand. Since the subdifferential of this integrand is defined on the whole effective domain, the problem of the validity of the Euler ...
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1991
This chapter is devoted to studying boundary value problems for second-order elliptic equations. The variational (also known as Hilbert space) approach to the Dirichlet problem is emphasized. Maximum principles are discussed in §5.10 and §5.11, which are independent of the preceding sections and are essential reading along with §5.1, §5.2, and §5.3 ...
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This chapter is devoted to studying boundary value problems for second-order elliptic equations. The variational (also known as Hilbert space) approach to the Dirichlet problem is emphasized. Maximum principles are discussed in §5.10 and §5.11, which are independent of the preceding sections and are essential reading along with §5.1, §5.2, and §5.3 ...
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