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We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxedminimizers of strongly quasiconvex functionals \begin{align*}\mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquadu\colon\Omega\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition$|F(z)|\leq c(1+|z|^{q})$, $z\in\mathbb{R}^{N\times n}$, and natural $p$-meancoercivity conditions on $F\in\mathrm{C}^{\infty}(\mathbb{R}^{N\times n})$ forthe basically optimal exponent range $1\leq p\leqq<\min\{\frac{np}{n-1},p+1\}$. With the $p$-mean coercivity condition beingstated in terms of a strong quasiconvexity condition on $F$, our resultsinclude pointwise $(p,q)$-growth conditions as special cases. Moreover, wedirectly allow for signed integrands which is natural in view of coercivityconsiderations and hence the direct method, but is novel in the study ofrelaxed problems.  In the particular case of classical pointwise $(p,q)$-growth conditions, ourresults extend the previously known exponent range from Schmidt's foundationalwork (Arch. Ration. Mech. Anal. 193, 311-337 (2009)) for non-negativeintegrands to the maximal range for which relaxations are meaningful, moreoverallowing for $p=1$. As further key novelties, our results apply to thecanonical class of signed integrands and do not rely in any way on measurerepresentations \`{a} la Fonseca & Mal\'{y} (Ann. Inst. Henri Poincar\'{e},Anal. Non Lin\'{e}aire 14, 309-338 (1997)).

Quasiconvex functionals of $(p,q)$-growth and the partial regularity of relaxed minimizers