We consider the nonlinear elliptic problem −div(a(x,Du)) + b(x, u) = μ with homogeneous boundary conditions in an open (possibly unbounded) subset of R^N , N >2. μ is a Radon measure with bounded variation, and u ->−div (a(x,Du)) +b(x, u) is a monotone operator acting in W^{1,p}_0, 1 < p < N. We prove that for every μ there exists at least a renormalized solution u to the problem, that is a distributional solution with additional summability properties. Moreover, if the operator is strictly monotone and μ does not charge sets of capacity zero, such a solution is unique.
Renormalized solutions to elliptic equations with measure data in unbounded domains / Malusa, Annalisa; Porzio, Maria Michaela. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - 67:8(2007), pp. 2370-2389. [10.1016/j.na.2006.09.007]
Renormalized solutions to elliptic equations with measure data in unbounded domains
MALUSA, ANNALISA;PORZIO, Maria Michaela
2007
Abstract
We consider the nonlinear elliptic problem −div(a(x,Du)) + b(x, u) = μ with homogeneous boundary conditions in an open (possibly unbounded) subset of R^N , N >2. μ is a Radon measure with bounded variation, and u ->−div (a(x,Du)) +b(x, u) is a monotone operator acting in W^{1,p}_0, 1 < p < N. We prove that for every μ there exists at least a renormalized solution u to the problem, that is a distributional solution with additional summability properties. Moreover, if the operator is strictly monotone and μ does not charge sets of capacity zero, such a solution is unique.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
