Abstract
We prove Lloc∞ estimates for positive solutions to the following degenerate second order partial differential equation of Kolmogorov type with measurable coefficients of the form ∑i,j=1m0∂xiaij(x,t)∂xju(x,t)+∑i,j=1Nbijxj∂xiu(x,t)−∂tu(x,t)++∑i=1m0bi(x,t)∂iu(x,t)−∑i=1m0∂xiai(x,t)u(x,t)+c(x,t)u(x,t)=0 where (x,t)=(x1,…,xN,t)=z is a point of RN+1, and 1≤m0≤N. (aij) is a uniformly positive symmetric matrix with bounded measurable coefficients, (bij) is a constant matrix. We apply the Moser’s iteration method to prove the local boundedness of the solution u under minimal integrability assumption on the coefficients.
Highlights
We consider second order partial differential operators of Kolmogorov-Fokker-Planck type of the formL u(x, t) := ∂xi aij(x, t)∂xj u(x, t) + bijxj ∂xiu(x, t) − ∂tu(x, t)++ bi(x, t)∂iu(x, t) − ∂xj (ai(x, t)u(x, t)) + c(x, t)u(x, t) = 0, i=1 i=1 (1.1)A(x, t) = (aij (x, t))1≤i,j≤N, where aij is the coefficient appearing in (1.1) for i, j = 1, . . . , m0, while aij ≡ 0 whenever i > m0 or j > m0
As the operator L is non degenerate with respect to the first m0 components of x, we introduce the notation
As far as it concerns degenerate operators, Wang and Zhang obtain in [23] the local boundedness and the Holder continuity for weak solutions to L u = 0 by assuming the condition b1, . . . , bm0 ∈ Lq(RN+1), with q = Q + 2
Summary
As far as it concerns degenerate operators, Wang and Zhang obtain in [23] the local boundedness and the Holder continuity for weak solutions to L u = 0 by assuming the condition b1, . We find that, for every Ω1 ⊂⊂ Ω2 ⊂⊂ Ω3 ⊂⊂ Ω, there exist a positive constant c1 b Lq(Ω), Ω1, Ω2 such that u L2α(Ω1)≤ c1 a Lq(Ω), b Lq(Ω), c Lq(Ω), Ω1, Ω2 Dm0 u L2(Ω2), and, by considering u as a test function, we obtain the following Caccioppoli inequality As far as it concerns the Moser’s iteration, the above inequalities are applied to a sequence of functions uk := upk , with pk → +∞, in order to obtain an L∞ loc bound for the solution u.
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