Naslov
Memory effects in homogenisation: Linear second-order equations

Autori
Antonić, Nenad

Izvornik
Archive for Rational Mechanics and Analysis (0003-9527) 125 (1993), 1; 1-24

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
memory effects; homogenisation

Sažetak
The theory of homogenisation treats the question: If the solutions u epsiv of the problems A epsiv u epsiv =f converge weakly to the function u 0, can an operator A 0 be found such that u 0 is a solution of the problem A 0 u 0=f, and is A 0 of the same type as A epsiv ? We study an example where the answer is negative. We take A epsiv $$: = - a^\varepsilon (t){; ; \text{; ; }; ; }; ; \partial _x^2 + b^\varepsilon (t){; ; \text{; ; }; ; }; ; \partial _x + c^\varepsilon (t)$$ and show that A 0 $$: = - a_{; ; {; ; \text{; ; eff}; ; }; ; }; ; (t){; ; \text{; ; }; ; }; ; \partial _x^2 + b_{; ; {; ; \text{; ; eff}; ; }; ; }; ; (t){; ; \text{; ; }; ; }; ; \partial _x + c_{; ; {; ; \text{; ; eff}; ; }; ; }; ; (t) + K *$$ is an integrodifferential operator. The expression for K is deduced under two different sets of assumptions — bounds in L 1 or L 2. The L 2 setting uses the Fourier transform and natural assumptions on the coefficients a epsiv , b epsiv and c epsiv — boundedness in L infin and uniform ellipticity. The answer in the L 1 setting is obtained only under additional assumptions, which seem to be unnecessary. Finally, the description of the memory term is given for a problem on a bounded interval, by using the eigenfunction expansion and a representation theorem for Nevanlinna functions. In one space dimension the equation $$ - a^\varepsilon (t){; ; \text{; ; }; ; }; ; \partial _x^2 u^\varepsilon (x, t) + b^\varepsilon (t){; ; \text{; ; }; ; }; ; \partial _x u^\varepsilon (x, t) + c^\varepsilon (t){; ; \text{; ; }; ; }; ; u^\varepsilon (x, t) = f(x, t)$$ is studied, while in several space dimensions – part x 2 is replaced by the Laplace operator – Delta. Computations are done in several examples.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

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