Study on symmetric positive systems

(i)我们研究了有界开集上的对称正系统，在边界矩阵上假设了以下条件。也就是说，在边界的余维数为1的嵌入子流形之外，边界矩阵的秩是常数。在这种情况下，我们找到了一个合适的权函数，该权函数在上述子流形之外为正，从而获得了边值问题解的先验估计。利用这一先验估计，证明了边值问题在法向上正则解的存在性。这对于非线性扰动的应用是非常重要的。利用这一先验估计和光滑解的存在性，我们成功地得到了参考嵌入子流形附近的弱解的性质，几个例子表明，这是非常清晰的。这些结果，就二维域而言，是相当令人满意的。如果我们把这个结果应用到所谓的Triconi方程上，我们就得到了解的唯一性的另一个证明。至于三维域，还有一种基本情况我们不能处理。（ii）在边界矩阵秩变化的子流形上边界矩阵为零的情况下，阐明了边值问题的结构。在沿子流形的膨胀流形上，我们可以用一个简单的权函数得到一个先验估计。利用这一先验估计在膨胀空间中证明了在法向上平滑解的存在性。应用我们在(i)中使用的相同方法，我们能够检查子流形附近弱解的行为。应用这一结果，我们得到了线性化MHD方程在某些边界条件下的先验估计。

(i) We have studied symmetric positive systems on a bounded open set assuming the following conditions on the boundary matrix. That is, outside a embedded submanifold of codimension 1 in the boundary, the rank of the boundary matrix is constant. In this case, we found a suitable weight function which is positive outside the above mentioned submanifold so that an a priori estimate for solutions to the boundary value problem is obtained. Using this a priori estimate, we have proved the existence of solution to the boundary value problem which is regular with respect to the normal direction. This is very important to applications to non-linear perturbations. With the aid of this a priori estimate and the existence of smooth solutions, we succeeded to get the behavior of weak solutions near the reference embedded submanifold, which is very sharp as several examples show. These results, as far as concerning two dimentional domains, are fairly satisfactory. If we apply this result to so called Triconi's equation, we get another proof of the uniqueness of solution. As for 3 dimentional domains, there remains one fundamental case which we cound not treat.(ii) In the case that the boundary matrix is zero on the submanifold where the rank of the boundary matrix changes, we clarified the structure of the boundary value problem. On the blown up manifold along the submanifold, we can get an a priori estimate with a simple weight function. Using this a priori estimate in the blown up space, we proved the existence of smooth solution even with respect to the normal direction. Applying the same method that we employed in (i), we are able to examine the behavior of weak solutions near the submanifold. Applying this result we obtained a priori estimate for the linealized MHD equation under some boundary condition which has not been treated before.