Study on symmetric positive systems

(I)在边界矩阵满足下列条件的情况下，我们研究了有界开集上的对称正系统。也就是说，在边界中余维为1的嵌入子流形之外，边界矩阵的秩为常数。在这种情况下，我们找到了一个在上述子流形外为正的合适的权函数，从而得到了边值问题解的先验估计。利用这个先验估计，我们证明了关于法线方向正则的边值问题解的存在性。这对于非线性摄动的应用是非常重要的。借助于这个先验估计和光滑解的存在性，我们成功地得到了在参考嵌入子流形附近的弱解的行为，这是非常尖锐的，如几个例子所示。这些结果，就二维区域而言，是相当令人满意的。如果我们将这一结果应用到所谓的Triconi方程，我们就得到了解的唯一性的另一个证明。对于三维区域，有一种基本情况是我们不能处理的。(Ii)在边界矩阵在子流形上为零的情况下，我们澄清了边值问题的结构。在沿子流形爆破的流形上，我们可以得到一个具有简单权函数的先验估计。利用这个在爆破空间中的先验估计，我们证明了即使相对于法线方向也存在光滑解。应用我们在(I)中使用的相同方法，我们能够检验子流形附近的弱解的行为。应用这一结果，我们得到了线性化MHD方程在一些以前没有处理过的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.