Elliptic Equations

Murray H. Protter and Hans F. Weinberger
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Murray H. Protter: University of California, Department of Mathematics
Hans F. Weinberger: University of Minnesota, Institute for Mathematics and its Applications

Chapter Chapter 2 in Maximum Principles in Differential Equations, 1984, pp 51-158 from Springer

Abstract: Abstract Let u(x1 x2,..., x n ) be a twice continuously differentiable function defined in a domain D in n-dimensional Euclidean space. The Laplace operator or Laplacian Δ is defined as $$ \Delta \equiv \frac{{{{\partial }^{2}}}}{{\partial x_{1}^{2}}} + \frac{{{{\partial }^{2}}}}{{\partial x_{2}^{2}}} + \cdots + \frac{{{{\partial }^{2}}}}{{\partial x_{n}^{2}}}. $$ . If the equation Δu = 0 is satisfied at each point of a domain D, we say that u is harmonic in D or, simply, that u is a harmonic function.

Keywords: Harmonic Function; Maximum Principle; Elliptic Equation; Laplace Operator; Harnack Inequality (search for similar items in EconPapers)
Date: 1984
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DOI: 10.1007/978-1-4612-5282-5_2

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