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We establish a unique stable solution to the Hamilton-Jacobi equation u t þ HðKðx; tÞ; u x Þ ¼ 0;x 2 ðÀ1; 1Þ; t 2 ½0; 1Þwith Lipschitz initial condition, where Kðx; tÞ is allowed to be discontinuous in the ðx; tÞ plane along a finite number of (possibly intersecting) curves parameterized by t:We assume that for fixed k; Hðk; pÞ is convex in p and lim p!AE1 j H ðk;pÞ p j ¼ 1: The solution is determined by showing that if K is made smooth by convolving K in the x direction with the standard mollifier, then the control theory representation of the viscosity solution to the resulting Hamilton-Jacobi equation must converge uniformly as the mollification decreases to a Lipschitz continuous solution with an explicit control theory representation. This also defines the unique stable solution to the corresponding scalar conservation law u t þ ðf ðKðx; tÞ; uÞÞ x ¼ 0;x 2 ðÀ1; 1Þ; t 2 ½0; 1Þwith K discontinuous. # 2002 Elsevier Science (USA)

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