In our reading we had the following maximum principle for elliptic PDE of degree 2: Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a solution of the linear Dirichlet task $$ (Lu)(x):=\sum_{i,j=1}^{n}a_{ij}u_{x_i x_j}+\sum_{i=1}^{n}b_i(x)u_{x_i}+c(x)u=f(x)\text{ in }\Omega,\\ u=\varphi\text{ on }\partial\Omega $$ Let the following conditions be given: $$ \Omega\subset\mathbb{R}^n\text{ a bounded }C^0-\text{ domain},\\L\text{ uniformly elliptic in }\Omega,\\\forall~i,j\in\left\{1,\ldots,n\right\}: a_{ij},b_i,c,f\in C(\overline{\Omega}),\\\forall~x\in\Omega~\forall~i,j\in\left\{1,\ldots,n\right\}: a_{ij}(x)=a_{ji}(x),\\\varphi\in C(\partial\Omega). $$ Additionally assume that $$ \forall~x\in\Omega: c(x)<0 $$ Then $$ \max_{x\in\overline{\Omega}}\left\{\lvert u(x)\rvert\right\}\leq\max\left\{\max_{x\in\partial\Omega}\left\{\lvert\varphi(x)\rvert\right\},\sup_{x\in\Omega}\left\{\frac{\lvert f(x)\rvert}{\lvert c(x)\rvert}\right\}\right\} $$

So far, so good. This was in our reading. Now there is a homework concerning this sentence above.

Now the task is the following:: Let $u\in C^2(\Omega)\cap C(\overline{\Omega})$ be a solution of the following Dirichlet-task (for a PDE in divergence form): $$ \sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}\left(a_{ij}(x)\frac{\partial u}{\partial x_j}\right)+\sum_{i=1}^{n}\frac{\partial}{\partial x_i}(b_i(x)u)+\sum_{i=1}^nc_i(x)\frac{\partial u}{\partial x_i}+d(x)u=f(x)\text{ in }\Omega,\\ u=\varphi\text{ on }\partial\Omega $$ assuming the following conditions: $$ \Omega\subset\mathbb{R}^n\text{ bounded }C^0-\text{ domain},\\\forall~i,j\in\left\{1,\ldots,n\right\}: a_{ij},b_i\in C^1(\overline{\Omega}),\\\forall~i\in\left\{1,\ldots,n\right\}: c_i,d,f\in C(\overline{\Omega}),\\\forall~x\in\Omega~\forall~i,j\in\left\{1,\ldots,n\right\}: a_{ij}(x)=a_{ji}(x),\\\forall~x\in\Omega: A(x):=(a_{ij})\text{ positive definite},\\\varphi\in C(\partial\Omega) $$ Formulate an analog statement to the statement above and find a sufficient condition for it.

My hitherto existing ideas are the following:

At first, I calculated resp. converted the given PDE in divergence form in a PDE that is in the "usual" form, getting a PDE that looks at least similar to the PDE that is given in the sentence. To do so, I merely calculated the derivatives resp. used the product rule. I got $$ \sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^2 u}{\partial x_i x_j}+\sum_{i=1}^{n}b_i(x)\frac{\partial u}{\partial x_i}+d(x)u+\sum_{i,j=1}^{n}\frac{\partial}{\partial x_i}(a_{ij}(x))\frac{\partial u}{\partial x_j}+\sum_{i=1}^{n}\frac{\partial}{\partial x_i}(b_i(x))u+\sum_{i=1}^{n}c_i(x)\frac{\partial u}{\partial x_i}=f(x) $$

What I noticed was, that one condition in the above sentence is, that the function before $u$ is negative for all $x\in\Omega$. Here, the function that stands before $u$ is $$ d(x)+\sum_{i=1}^{n}\frac{\partial}{\partial x_i}(b_i(x)), $$ so my thought was, that it is necessary that $$ d(x)+\sum_{i=1}^{n}\frac{\partial}{\partial x_i}(b_i(x))<0\\\Leftrightarrow d(x)<-\sum_{i=1}^{n}\frac{\partial}{\partial x_i}(b_i(x)). $$

But I do not know if this is already enough to be able to apply the above sentence resp. to transfer the sentence on the homework problem.

Can anybody help resp. explain to me, if I got the sense of this homework and if there are other things that must be fullfilled?

Sincerely yours


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