Wyniki wyszukiwania

1

The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation

100%

Watson N. A.

Annales Polonici Mathematici

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2000

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tom 75

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nr 3

281-287

EN

Let L be a second order, linear, parabolic partial differential operator, with bounded Hölder continuous coefficients, defined on the closure of the strip $X = ℝ^{n} × ]0,a[$. We prove a representation theorem for an arbitrary $C^{2,1}$ function, in terms of the fundamental solution of the equation Lu=0. Such a theorem was proved in an earlier paper for a parabolic operator in divergence form with $C^{∞}$ coefficients, but here much weaker conditions suffice. Some consequences of the representation theorem, for the solutions of Lu=0, are also presented.

2

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Boundary integral representations of second derivatives in shape optimization

88%

Eppler K.

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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2000

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tom 20

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nr 1

63-78

EN

For a shape optimization problem second derivatives are investigated, obtained by a special approach for the description of the boundary variation and the use of a potential ansatz for the state. The natural embedding of the problem in a Banach space allows the application of a standard differential calculus in order to get second derivatives by a straight forward "repetition of differentiation". Moreover, by using boundary value characerizations for more regular data, a complete boundary integral representation of the second derivative of the objective is possible. Basing on this, one easily obtains that the second derivative contains only normal components for stationary domains, i.e. for domains, satisfying the first order necessary condition for a free optimum. Moreover, the nature of the second derivative is discussed, which is helpful for the investigation of sufficient optimality conditions.

3

Existence of the fundamental solution of a second order evolution equation

75%

Bochenek J.

Annales Polonici Mathematici

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1997

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tom 66

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nr 1

15-35

EN

We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence ${A_n(t)}$ of bounded operators.

Ograniczanie wyników

2 Annales Polonici Mathematici

1 Discussiones Mathematicae, Differential Inclusions, Control and Optimization

1 Bochenek J.

1 Eppler K.

1 Watson N. A.

2 2000

1 1997

The representation of smooth functions in terms of the fundamental solution of a linear parabolic equation

Boundary integral representations of second derivatives in shape optimization

Existence of the fundamental solution of a second order evolution equation