# Existence of the degenerate elliptic PDE coefficient condition

This is a question related to the theory presented in a book on degenerate elliptic PDEs. The book builds a theory for equations of the form: $$\sum a_{ij}u_{x_ix_j}+\sum b_{i}u_{x_i}+cu=0$$ with $A=a_{i,j}$ is a positive semi-definite matrix. Then, the theory is provided. On the existence, one of the conditions for the coefficients, along with considering Holder continuity for them, the author puts the constraint for the coefficient $c\leq c_0<0$. This looks very odd to me as consider $c=const$, I can always eliminate that part of the equation by multiplying by the exponential with the constant $c<0$ and obtain a new equation of the form without the term $cu$!(the proof of that theorem is based on the method of elliptic regularization, i.e. add some diffusion $\sum \epsilon u_{xx}$ to get a strongly parabolic case and apply the regular theory, then pass to a limit) So, why is that condition present in that form? Then, for the uniqueness, the similar condition is provided. To be precise, one can write the adjoint operator $L^{\ast}$ of the form: $$L^{\ast}u=\sum a^{\ast}_{ij}u_{x_ix_j}+\sum b^{\ast}_{i}u_{x_i}+c^{\ast}u=0$$ and for uniqueness it requires a condition $c^{\ast}<0$. I wish I could make a copy of the pages but I would appriciate any intuition where this condition is coming from and it can't be eliminated.

• In general, this condition is imposed to obtain coercivity. Anyway, I do not understand your method, can you explain it better? – Tomás Mar 11 '13 at 13:07
• But, coercivity is the condition for $a$, or $c$ would affect that as well? I can refer to the book I am reading: Olejnik and Radkevic: Second order equations with nonnegative characteristic form. Paragraph 5. Theorem on existence, I did not get into the proof really, the idea is to do the elliptic regularization, that is add $\epsilon \sum u_{x_ix_j}$ and then use the regular elliptic theory for strongly elliptic equations with $\tilde{a}\geq a_0>0$. The book considers the more general $c=c(x)$, but we can just think of it as a constant to put some contraints on it in order to have existence. – Medan Mar 11 '13 at 14:10
• Take a look into my answer to this question (after equation (2)): math.stackexchange.com/questions/285013/… – Tomás Mar 11 '13 at 14:19
• Note that I am not saying that there isn't a solution to the problem if the constraint is not assumed, but in general you lose coerciveness and problably you not gonna look for minimum, instead you have to look for saddles (if it exists). – Tomás Mar 11 '13 at 14:22
• Dear @medan, if you look into my answer, you will see that the case treated there is: $a_{ij}=-\delta_{ij}$, $b_i=0$ and $c \leq\frac{\lambda_1}{2}$. This is related to your problem because it is your problem with those conditions. NOte that if $\lambda>\frac{\lambda_1}{2}$, you dont have coercivity, so the fator $c$ in fact affects the coerciveness of the operator. – Tomás Mar 11 '13 at 17:38